Portfolio - aidan meacham | cello - physics

Aidan Meacham - Portfolio
Contents
Musicology
2. Classical Samples in Rap
18. Glass & Herrmann: Divergent Film Composers
31. Links to Senior Recital
Math / Physics
32. Linear Algebra for Image Compression
44. Image Compression Presentation
75. ESPI Image (example)
76. ESPI Laser Paths (Illustrated)
77. ESPI Setup
Design
78. Crosscurrents Deadline Extension Poster, Spring 2013
79. Crosscurrents Deadline Extension Poster, Spring 2014
80. Crosscurrents Submission Deadline Poster, Spring 2014
81. ASUPS Poster, Fubuki Daiko
82. ASUPS Poster, La Familia Valera Miranda
83. Junior Recital Poster (2013)
84. Senior Recital Poster (2014)
85. Resume
493 Term Paper
Aidan Meacham
Classical Samples in Rap: A Study
The musical practice of sampling, where a recording is repurposed in a new
context, was pioneered by minimalists in the late 1960s, was popularized in the 1970s by
DJs experimenting with electronic music, and had perhaps the greatest influence on the
development of hip-hop. Surrounded by issues including copyright, authenticity, and the
development of technology, sampling is in many ways an extension of the practice of
quotation in classical music, which encompasses many of the same issues. Particularly in
terms of hip-hop, sampling and rhyming are two of the most significant aspects of the
genre, and each can hold cultural meaning and in combination can add a level of
commentary to a track. The ability to juxtapose symphonic music, with its cultural and
economic connotations, with expletive and excess-laden lyrics is a unique and powerful
tool in the hands of producers that use its syncretic power to create new dialogues. This
paper examines samples of classical (or classical-sounding) music for beats in hip-hop,
and attempts to explain the reasons for its use, specifically in three stylistic contexts: oneupping, messages of uplift, and musical transformation for word painting.
In order to understand the cultural context of sampling, it is important to
understand the history of the practice, and this begins with the technology that allowed it.
In 1964, Steve Reich's composition "It's Gonna Rain" was created as a result of his
accidental "invention" of phase shifting - when listening to two identical loops (repeating
a preacher's exhortation, the titular "It's Gonna Rain") on two separate tape recorders, the
tracks went out of sync, creating a phase shift. 1 This is one of the first examples of
"sampling" in a musical sense - the practice of musique concrète had utilized recordings
in a repurposed setting, but Reich was the first to isolate a loop and repeat it, emphasizing
the inherent rhythms of the recording. The audio loops that Reich created can be seen as a
precursor to beats in hip-hop music, where a repeated "break" that is isolated from a
recording becomes the basis for an entirely new track.
Joseph Schloss, in Making Beats: the Art of Sample-Based Hip-Hop, points out
that party DJs originally used two turntables and a mixer to prepare a record on one
turntable "while another was still playing, thus allowing for an uninterrupted flow of
music," eliminating the stop between tracks. 2 Much like Reich's loops and phase shifting,
Schloss posits that "the central innovation of early hip-hop was the use of this system
with two copies of the same record for various effects, particularly the isolation of the
'break.'" David Toop asserts that the isolation of breaks was motivated by the dance
aspect of parties, as DJs would just "[play] the fragments that were popular with the
dancers and [ignore] the rest of the track." 3 In the late 1970s, hip-hop as a genre was born
as a result of break-based music becoming completely divorced from the original
recordings. 4 Sampling grew out of this tradition in the 1980s, with the advent of digital
tools that could sample, modify, and sequence materials from a variety of records,
creating the ability to completely produce a track from pre-existing sources. 5 According
to Andrew Bartlett, "the hip hop archive serves as a miniaturized repository for vast
1
"Steve Reich - Early Tape Pieces," last modified April 2000, http://www.furious.com/perfect/ohm/reich2.html.
Joseph Schloss, Making Beats: The Art of Sample-Based Hip-Hop (Middletown: Wesleyan University Press,
2004), 31.
3
David Toop, Rap Attack 2 (New York: Serpent's Tail, 1991), 60.
4
Schloss, 33.
5
Schloss, 36.
2
interactive historical material – interactive because all archival material is handled by the
archivist, who listens carefully [...] for the beats and snippets which will accompany and
be accompanied by vocalized narrative." 6
As a side note, "one effect of this approach is hip-hop's celebration, almost unique
in African American music, of the solitary genius," Schloss specifies. "Hip-hop producers
hold an image of themselves that recalls nothing so much as European art composers: the
isolated artist working to develop his or her music." 7 Schloss additionally comments that
the advent of digital sampling in the modern era has made it even easier for a producer to
create a track from beginning to end by themselves, controlling every aspect of a track's
production. This, in turn, allows one producer to efficiently make a point with or signify
upon the samples they choose to utilize in their music, working with a rapper to create a
synthesis of sound and lyric at a level of unprecedented incorporation. This is tempered,
however, by the consideration that producers are sometimes not looking to make a point,
but just want to make a track that sounds the best and is in line with their aesthetic vision
and cultural values (such as the "rareness" of a particular cut or sample, for example).
The messages we will analyze are those on tracks that utilize classical music as a
primary sample, and can be generally categorized into three types. I am making these
generalizations not to place the raps or beats into limiting boxes but to examine the
motivation or ideals behind using samples of classical music semiotically, musically, or
culturally. The three categories, as mentioned earlier, are one-upping, uplift, and
transformation / word painting, and each comes with their own motivations for producing
a track that reflects these general attributes. A track that is categorized as "one-upping" is
6
Andrew Bartlett, "Airshafts, Loudspeakers, and the Hip-Hop Sample" in That's the Joint!, ed. Murray
Forman et al. (New York: Routledge, 2012), 573.
7
Schloss, 42
the practice of extolling one's attributes that are superior to others, whether musical (in
terms of rapping or production), material (money and power), or lifestyle (being a
gangster, player, etc.). Word painting, a term usually associated with classical music (but
applicable to any music), refers to a work with text where the music is treated
referentially to the words being spoken or sung (to some degree); for example, a melody
that descends on the word "down" or accompaniment that sounds flowing when singing
about a river. Tracks in this category engage with the music of the sample and often have
extreme transformations of the music they utilize in order to better fit the text (or write
text to reflect the nature of the sample; regardless, these tracks are concerned with
music). Tracks we will categorize as having a message of uplift frequently include lyrics
that urge listeners to change their lifestyle for the better. Each of these generalized
categories utilize the classical samples that form the basis of the track differently, and
through analysis of individual songs, the purpose behind their usage will become clear.
It is worth noting that in many of the tracks discussed below the rapper is not
necessarily the producer. However, in terms of analysis, we will treat the pair (or group,
as the case may be) as contributing equally to the overall concept of the track, and as
such, the rapper's name will be used interchangeably for both the individual or the rapperproducer team. Analysis of the practical aspects of production would be more concerned
with these issues, but in considering the wider cultural meaning and context (or lack
thereof) created by the use of classical samples, minimizing these issues will keep the
focus on the more artistic interplay of lyrics and beats that provides the evidence for
arguments about motivation for using classical music in each context. In addition, many
lyrics are not released or verified by the artist, so the versions I have included here are as
complete and accurate (to my ear) as possible. I have not made any effort to reduce,
replace, or censor any expletives or potentially offensive lyrics, as this would
compromise their accuracy and authenticity, though I do recognize the sensitivity of the
subject and have done my best to portray the lyrics as consistently and neutrally as
possible to preserve the artists' intentions.
One-upping (known as braggadocio) in rap music is believed to come from the
early competitiveness of freestyling MCs who tried to outdo each other in terms of
technical rapping skill, but eventually evolved into explicit bragging with regards to
money, fame, or other aspects of lifestyle and achievement. This development can be
seen as a reflection of the African-American cultural practice of "playing the dozens,"
where insults are traded (often in rhyme) between two competing players. 8 This is
especially important for the perception of a "hard" gangster, according to Greg
Dimitriadis, "embodies such capitalist values as rugged individualism, rampant
materialism, strength through physical force, and male domination, while he rejects the
very legal structures which define that culture." 9 Recorded albums sometimes contain
tracks that are manifestations of real life feuds between rappers; often, these tracks
contain braggadocio or insults of the other rapper, and can result in a response in another
album, continuing the feud. Now, though still an important skill for a rapper, braggadocio
has become a trope in popular culture often used to make fun of "gangsta rap," a genre
born in the 1980s when many rappers had ties to gangs. In the 1990-2000s, the genre
became extremely lucrative and mainstream, causing some to believe that many new
8
Harry Lefever, "'Playing the Dozens:' A Mechanism for Social Control," Phylon 42 (1981): 73.
Greg Dimitriadis, "Hip-Hop: From Live Performance to Mediated Narrative" in That's the Joint!, ed.
Murray Forman et al. (New York: Routledge, 2012), 589.
9
gangsta rappers were fake – a record company's attempt to sell the rapper lifestyle and
garner media attention (thus increasing sales) through depictions of violence and drugs.
One example of this style is "Hate Me Now" by Nas, who raps with Puff Daddy
over a beat based off Carl Orff's Carmina Burana. A grandiose introduction in a slow
tempo that utilizes the well-known "O Fortuna" sets up the hook along with the main beat
of the song, a loop taken from "O Fortuna" with faster, syncopated drums. This first hook
is followed by a vocal sample from Orff, then proceeds into the first verse. Nas raps:
Don't hate me, hate the money I see, clothes that I buy
Ice that I wear, flows that I try, close your eyes
Picture me rollin', sixes, money foldin'
Bitches, honeys that swollen to riches, Nas get in ya
Most critically acclaimed Pulitzer Prize winner
Best storyteller, thug narrator, my style's greater
Model dater, big threat to a lot of you haters
Commentators ringside try watchin' my paper
As Nas continues, he extols his authenticity, relating the envy of those who would even
try to kill him to get the things he has. The use of Carmina Burana is motivated twofold:
the sonic qualities of the Orff fit the grandiose feel of the track, and the minor key
expresses his anger with those who hate him for his riches. The hook, "You can hate me
now / but I won't stop now / cause I can't stop now / you can hate me now" makes clear
the idea that it is his money and fame people hate him for, not who he is as a person. In
this regard, the usage of classical music is primarily for the effect of its perceived
grandeur, authenticity, and associations with wealth: since it sounds epic and is authentic
classical music, Nas put his own stamp on it in order to claim the biggest sound and
authenticity for himself.
A second example of this categorization is Ludacris' "Coming 2 America," where
a sample of the 4th movement of Dvorak's 9th symphony is the basis for the beat. After a
short introduction spoken over a beat sampling the Dies Irae from Mozart's Requiem, the
main Dvorak sample enters, continuing during the rest of Ludacris's raps, with Dies Irae
appearing during spoken interludes. An example of Ludacris's braggadocio is as follows:
Fuck you too! What you wanna do, scrawny nigga
But I got an arsenal of automatics down to twenty-twos
Know how to use 'em, fight dirty as shit
I throw a grenade and all-in-one bury a clique
You see y'all got it all wrong like women in tuxedos
And comin' up shorter than five Danny DeVitos
I'm on a cool ranch, get laid more than Fritos
With five strippers, four wives and three amigos
Like Nas, Ludacris utilizes the two "epic sounding" samples, with full symphonies and
choirs in order to illustrate how rich and hardcore he is. By sampling a symphony and
choir, Nas and Ludacris are essentially arguing that they are "classic" like the symphonic
repertoire, as well as making an economic claim about the funds necessary to hire a full
symphony. Despite the fact that the sound we hear is a sample of a previous recording,
the audiological cue is all that is necessary to illustrate the depth of their pockets, taste,
and skill – in a way, the metaphorical appropriation of the sample's "classicalness" is
parallel to the literal use of the sample.
One final example of musical one-upmanship is Three 6 Mafia's "Dangerous
Posse." Similarly to Nas and Ludacris, the members of Three 6 Mafia overlay their more
violence-oriented lyrics with epic sounding strings; however, in this case, the main
sample is from the soundtrack to the 1996 film Romeo + Juliet, not a canonical classical
piece. In addition, the "classicalness" of the sample is obscured by the bassline and beats
overlaid on the loop. These characteristics imply that, opposed to Nas and Ludacris'
tracks, Three 6 Mafia picked the sample for its sonic qualities alone, whereas the
treatment of the samples in Nas and Ludacris' tracks imply that they wanted to be
associated with the "classicalness" of their samples. In other words, Three 6 Mafia only
wanted the sound, whereas Nas and Ludacris wanted the sound and the elitist values of
classical music.
A second category of tracks featuring classical music is that of word painting and
extreme transformation to fit a text or particular aesthetic ideal. These tracks are often the
most sonically distant in comparison to the original piece being sampled, but engage in
drastic transformation as an artistic decision that is more deliberate, modern, and possibly
more difficult than more straightforward examples of sampling. This category is more
fluid than the others, and could contain many of the tracks in other categories as well, but
the samples contained therein are often the most complex, and placed within the most
complex textures, making their analysis more fulfilling than a more straightforward
quotation. In particular, these tracks engage with the content of the music (as opposed to
its connotations) and, in their transformation of the samples themselves, illustrate texts.
The Art of Noise's 1999 concept album entitled The Seduction of Claude Debussy
aimed to be a metaphorical chronicle of Debussy's life, intermixing his music with hiphop, jazz, and spoken word. The album (which could be identified as belonging to the
"new-age dance" genre) is primarily instrumental with occasional vocals, but the track
"Metaforce" contains a rap by Rakim, an early and extremely influential rapper. The track
begins with a Debussy-esque piano solo with heavy reverb, then jumps into an
electronica influenced rap beat that utilizes synthesized strings on the same harmonic
progression as the piano solo. Rakim enters, stating "This is a metaphor" early on, then
continues,
I leave time suspended and break gravity’s law
Metaforce to the world ain’t spinning no more
And from there I put sounds to hear, no order there
So that we’re something y’all will compare to Baudelaire
Portions of the rap illustrate perfumed French gardens, "the evening air in summertime,"
and other poetic subjects, all over the floaty-sounding beat. The clear intention is to take
Debussy's music that sounds weightless and apply a metaphorical lyric in order to create
a synthesis of old and new. This distinct transformation of the classical samples is
founded in a conceptual manner, and as such, can be viewed in the categorization of
word-painting, albeit in a manner that reverses the typical process – in this case, Rakim's
lyrics match the weightlessness of Debussy's music, not the other way around.
One example where a sample of a classical work is transformed to create a new
aesthetic is the track "Same Old Thing" by Manchester-based The Streets. The samples
heard throughout are also taken from Dvorak's 9th Symphony, but the main repeated
sample is dramatically transformed. The loop formed by this sample is purposefully cut
in a way where the beats of the sample come uncomfortably late in comparison with the
steady beat that overlays the track, giving an unpolished, stumbling feel to the music.
When examining the lyrics, it becomes clear that this is intentional – the stream-ofconsciousness style rap details a night at the bar, devolving into drunken debauchery, and
the listener realizes that the music is reflective of this drunken state as well. The lyrics
have a sense of the everyday and the utilitarian so it is interesting that The Streets
chooses to utilize Dvorak for their beat, as he was one of the first composers to seek out
folk (everyday) materials to use in classical composition. This is one example where the
transformation of the classical sample is brilliantly executed in order to serve the story of
the track, and can have a greater significance in its origin than sonic variety or quality.
Another example of this style is Cunninglynguists' introduction track to their
debut album Will Rap for Food, entitled "Lynguistics." The examples of artists' one-
upmanship we have examined utilized epic samples that bolstered their credibility, but
like The Streets, Cunninglynguists utilizes a sample of classical music in transformation
to show off their skills. There is some explicit bragging that occurs, but the lyrical and
musical excellence is really how Cunninglynguists are arguing for their legitimacy. They
sample the Tchaikovsky Violin Concerto, Mv. 1, but layer on additional sonic materials,
phasing between the two to create an impressive, varying soundscape. The lyrics are
complex, as well, with multi-layered meanings and pop culture references. In this case, a
parallel is being drawn between the virtuosity implicit in the Tchaikovsky with their own
virtuosity as rappers and producers. While there is no specific conceptual connection
between the sample and the track, the blending of classical music with their own, akin to
Nas and Ludacris, is meant to be indicative of their skills and taste.
Many of the one-upping tracks analyzed above express a tendency toward
violence and materialism that some rappers speak out against. These rappers created
tracks that are oriented toward uplifting their audience socially and economically, and
can therefore be characterized as containing a message of uplift. Sometimes these tracks
are aimed at other artists, and some tracks attempt to connect with a base of youth to
emphasize upward mobility and avoiding derogatory stereotypes. In addition, "uplift"
tracks can be either sentimental or progressive in nature, but generally, the messages
communicated promote non-violence, empathy, and upward mobility. The classical
samples used in this category are typically less grandiose and more introspective than in
braggadocio, as even a cursory listen will show, pointing to the nature of the tracks.
Coolio's "C U When U Get There" is an example of a positive message being
reinforced with a sample of classical music – in this case, Pachelbel's Canon in D. The
lyrics reflect on violence and greed, urging others to forgo them in search of brotherhood.
One lyric in particular at the end of the track emphasizes this message:
As we walk down the road of our destiny
and the time comes to choose which it gonna be
the wide and crooked or the straight and narrow
we got one voice to give and one life to live
stand up for something or lie down in your game
listen to the song that we sing
it's up to you to make it be
I guess I'll see you when you see me
The title's meaning is twofold in terms of what "there" is, referring both to a non-violent
lifestyle free of motivation by greed, and alternatively, heaven, where victims of violence
have gone and are waiting for those who will follow. The lyric in the chorus "if you ever
get there" reinforces this interpretation, as one could either fail to adopt the
aforementioned mindset or not make it into heaven. The chorus is sung over the sample
of Pachelbel's canon, and is a choral setting, which emphasizes the nostalgic aspects of
the track as well as contributing to its classical and gospel feel. In doing so, Coolio is
implying religion and traditional morality in order to reinforce his message to listeners.
A second example of a track that utilizes classical music to promote a message is
Xzibit's "Paparazzi," which grapples with the issue of fame in the rap business, and the
lengths rappers will go to prove themselves to be authentic and hardcore, even if it is only
an act. The track has the feel of a lament due to the use of Fauré's Pavane, Op. 50, with a
solo singer layered with a beat, at times using effects like dramatic reverb and
equalization to drive home the atmosphere. Xzibit laments the perceived need for a rap
artist to be gangster, instead emphasizing authenticity and moderation.
So ease off the trigga talk, you ain't killin' shit
it's not affecting me or the niggaz that I'm chillin' with
I don't believe the hype or buy a woof ticket
Nigga you make a gang of noise and never seem like a cricket
I guess that's why we never kick it
a lot of niggaz are soft and get tossed tryin' to fuck with the liquid
How many niggaz do you know like this?
Also claimin' that they're riding but they really turn bitch
It don't make sense
Either you're a soldier from the start
or an actor with a record deal tryin' to play the part
This proceeds directly into the chorus, which repeats "It's a shame / niggaz in the rap
game / for the money and fame," reemphasizing that some rappers' pursuit is only
material, when an authentic rapper actually has something to say and has been working at
it since before they got a contract. In emphasizing his own bootstrapping beginnings,
Xzibit argues that his authenticity and moderation is superior to new rappers who are
commercial stars and rap about violence and materialism to fulfill what they (or record
company executives) see as the rapper lifestyle. In doing so, Xzibit simultaneously calls
into question the authenticity of new gangsta rappers and exhorts listeners to stay true.
One final example of a serious track that sends a positive message is Nas' "I Can,"
which uses the beginning of Beethoven's Für Elise as its main sample as well as a
metaphorical tool for its message. The hook, which appears over the first full statement of
the sample, is sung by kids and is meant to affirm black youth's upward mobility:
I know I can
Be what I wanna be
If I work hard at it
I'll be where I wanna be
The significance of the Beethoven sample lies in that Für Elise is frequently a piece used
to teach beginners, frequently youths, to play piano. This leads to implications of
practicing and dedication, the theme of the track. This is reinforced by the first
appearance of the sample; when the first break is heard, the piano sounds out of key in
comparison, but eventually falls into place. As the track goes on, more and more is built
on top of this sample, giving the impression that its repetition (practice) has improved it
more and more until the final chorus when it is joined by strings, implying the practice of
a number of peers and their cooperation to make music. Because the metaphor of the
sample is the primary motivating factor, not the musical content therein, "I Can" falls
firmly into the "uplift" category, though the subtle transformation (through layering) of
the sample hints at the overlap between categories, as seen before with "Lynguistics," for
example. All three verses address upward mobility, relating through anecdote and
storytelling the cautions youth should exercise in avoiding drugs and violence and
affirming their capability to become productive members of society if they work for it.
A significantly less serious track that still expresses some of the same sentiments
as the above is D'Mite's "Read A Book," which utilizes a loop from Beethoven's Fifth
Symphony and tropes common to rap while criticizing the lifestyle choices rap often
encourages. It is important to note that, in comparison to the other tracks presented here,
this was released on MySpace, not on a commercially-produced album, where the aspects
of satire and unoriginality would've been frowned upon by record executives who are
trying to sell albums of commercialized and purposefully excessive "gangsta" rap. The
satirical nature of the song is apparent from the beginning, with a Lil' Jon style "yeah,"
followed by a spoken intro over a piano reduction of the first theme of Beethoven's Fifth.
Wassup y'all, ha ha ha
This your boy D'Mite
You see I usually do songs with like
hooks and concepts and shit, right?
But fuck that man, I'm tryin' to go platinum
So I'm 'bout to go rock this shit
Check this out y'all, uhh
As D'mite continues into the chorus, the sample loops for the rest of the track, and most
of the raps are in the form of the chorus, but with differing words. The lyrics are always
presented with a satire of some aspect of both rap culture and its music, with frequent
interjections in the style of Lil Jon, explicit language, and chorus-style yelled lyrics.
D'Mite forcibly exhorts his listeners to, among other things, read a book (cleverly set, at
one point, in a spelled out manner: "R-E-A-D A B-O-OKAY"), raise their kids, drink
water (instead of alcohol, as encouraged by other rappers), buy property, stop spinning
rims (a jibe at both idleness and materialism), and observe personal hygiene.
While obviously satirical in nature, the track effectively presents these issues in a
direct manner, and its use of Beethoven is clearly intended to be part of the track's
emphasis on the arts. D'Mite encourages his listeners to take up Beethoven alongside rap
beats through its inclusion on the track. The dramatic nature of the sample contributes to
this intention, as it is made out to be a rap beat as legitimate as any other. Despite its
lampooning nature, "Read a Book" still carries a positive message, and its usage of a
classical sample reinforces the artist's belief that youth participation in the arts is
important and more desirable than their participation in rap culture.
While it would be impossible to discuss the motivations of the use of classical
samples in general (other than the commonality that a producer thought it sounded good),
the three categories above have moderately consistent intentions for their inclusion of
classical music. For each, the trends found reveal the aspirations and objectives of the
artists in each respective category. Tracks that sample classical music in service of
braggadocio primarily include epic sounding symphonic and operatic music in order to
claim the larger-than-life status of the sound for themselves, in terms of what Joanna
Demers considers the "elitist values" of classical music: "wealth, intelligence, [and]
refined taste." 10 Tracks that drastically transform a classical sample are frequently in
service of a larger concept or are reflective of the text, much like the classical practice of
10
Joanna Demers, Steal This Music (Athens: University of Georgia Press, 2006). 43.
word painting, although in some cases, the direction of "painting" is reversed. In these
cases, the use of classical is often in line with the aesthetic the artists are trying to evoke,
and the virtuosity or complexity of the classical music and production thereof are highly
desirable qualities. The final category, uplift, frequently uses samples of classical music
to imply cultural values associated with classical music like practice and education, and
can also represent a rejection of rap culture.
Though the sonic qualities of a sample are a major consideration for inclusion in a
track, these idiomatic uses of classical samples point to a more compelling reason for
their inclusion. What is interesting about these samples regardless of category, as a
cursory review of the analyses above reveals, is that tracks rarely engage with the race of
classical music or the nature of its establishments through sampling, but rather import its
cultural associations and values – in other words, artists are not typically concerned with
racial signifyin(g) on classical music. Hip-hop is a distinctively African-American genre,
and in a stark contrast to attempts at racial uplift through black classical music during the
Harlem Renaissance, these artists are not trying to create "concert works" through their
use of classical samples, nor are they even commenting on issues of race through
classical music. Rather, artists use classical connotations as a tool for a new brand of
uplift and commentary, where the purpose behind its usage is driven by its longestablished connotations, musicality and flexibility. In creating fresh dialogues through
the appropriation of old materials, whether self-serving glorification or uplifting
principles, rappers have given a new beat to classical works.
Works Cited
Bartlett, Andrew. "Airshafts, Loudspeakers, and the Hip-Hop Sample" in That's the
Joint!, ed. Murray Forman et al. New York: Routledge, 2012.
Demers, Joanna. Steal This Music. Athens: University of Georgia Press, 2006.
Dimitriadis, Greg. "Hip-Hop: From Live Performance to Mediated Narrative" in That's
the Joint!, ed. Murray Forman et al. New York: Routledge, 2012.
Lefever, Harry. "'Playing the Dozens:' A Mechanism for Social Control," Phylon 42
(1981): 73-85.
OHM. "Steve Reich - Early Tape Pieces," last modified April 2000,
http://www.furious.com/perfect/ohm/reich2.html.
Schloss, Joseph. Making Beats: The Art of Sample-Based Hip-Hop. Middletown:
Wesleyan University Press, 2004.
Toop, David. Rap Attack 2. New York: Serpent's Tail, 1991.
333 Term Paper
Aidan Meacham
Glass & Herrmann: Divergent Film Composers
When it comes to film, there are few techniques with the power to electrify,
pacify or terrorize an audience that compare to the musical score. The best directors and
film composers meld sight and sound into a harmonious whole, where music can have
subtle implications for a plot or drive home a brutal musical depiction of the action on
screen. Some of the most famous moments in cinema are remembered by their musical
underpinnings: what would Psycho's shower scene be without shrieking violins, or Jaws
without the terrifying bass undulations? Film music has the ability to play the audience as
well as any actor or cinematographer, and depending on the type of music incorporated,
can drastically alter the feeling, subtext, and action of a scene.
In this paper, I will focus on two composers and how their music interacts with a
film and the audience, and in particular, how the subtle differences between their musical
genres play into very different conceptual approaches to the treatment of thematic ideas,
both in terms of the music itself as well as within the text of the films they accompany.
Bernard Herrmann, most famed for his work in Alfred Hitchcock's films (though also a
classical composer), is sometimes referred to as proto-minimalist due to frequent use of
ostinato in his scoring (a side effect, perhaps, of the brutal insistence of rhythm in scenes
like Psycho's shower murder), whereas Philip Glass's ventures into film scoring utilize
the consciously minimalist idioms developed in his earlier classical works. Minimalism
refers to music that is subject to incessant repetition, where slight changes in musical
texture are made gradually in time so that the focus of the music is not the texture itself,
but rather, the transparency of the progression. Taken strictly in this fashion, all of the
scenes we will examine could fit this definition of minimalism; however, fundamental
differences in the purpose of each scene hint at differences in musical quality that bear
closer consideration. By comparing particular scenes in North By Northwest and Vertigo,
scored by Bernard Herrmann, with scenes in Koyaanisqatsi and The Hours, scored by
Philip Glass, the differences in cinematographic purpose and interpretation will
underscore the distinction of each composer's musical idiom and the importance of
making the subtle delineation between the two. In order to compare how Herrmann's
scores differ from minimalism, we shall first examine films scored by Glass. Since Glass
writes both classical works and film scores in a similar idiom, we can focus on his music
as the prototypical example of minimalist music, the purpose for its use in film, and its
overall thematic argument, to which we will subsequently compare to Herrmann's scores.
As one of the earliest examples of minimalism in film scores, Godfrey Reggio's
1983 film Koyaanisqatsi is also the most straightforward use of the idiom that we will
explore and, subsequently, will act as the ruler by which we will measure the
"minimalism" of the other scores. Koyaanisqatsi is a hopi word which translates to "life
out of balance," or interpretively, "a state of life that calls for another way of living." 11
The film itself is entirely comprised of images paired with non-diegetic music scored by
Glass, with no dialogue or text except for the titular definition presented at the end of the
film, so that the interplay between the screen and music is the main focus of the work,
leaving the audience to determine their own conception of the narrative. Being one of the
first films to utilize minimalist music, the score itself is relatively similar to other
11
Koyaanisqatsi, directed by Godfrey Reggio (1983; Santa Monica: MGM Home Entertainment, 2002),
DVD.
contemporary compositions by Glass (for example, 1982's Glassworks); however, when
placed in the context of Koyaanisqatsi, the musical tropes listeners commonly associate
with the classical genre take on a new meaning.
The most significant section of the film is accompanied by a track entitled "The
Grid," a visual and musical conflagration of technology with humans, presenting the
people in the film not as individuals, but as cogs in a machine. Rebecca Eaton, author of a
2008 doctoral dissertation on minimalism in film scores, argues that "the humans in these
scenes are not presented as subjects, with individual feelings and purposes. They do not
speak [...] or show emotion on their faces, they simply move, just like (and with) the
machines, as if programmed automatons." 12 By juxtaposing time-lapse and slow-motion
shots of automated machines and people performing repetitive tasks, and eventually
showing the two together (in a production line of cars for example), Reggio implies the
machine-like routines of every day life. Eaton points out that the music supports this
interpretation, with little or no differentiation in music between shots of human and
machine, 13 and quoting Alex Ross, points out that the score's difficulty requires a kind of
mechanistic virtuosity from its players: "Glass and his musicians become manic
machines, firing off notes like so many 0s and 1s." 14 15 In addition, the repetition of the
music itself and the use of the choir (the only human sound in the film) solely on
vocables is used as evidence for the roboticism of humanity (or at the very least,
commonality or lack of individualism), a theme that is now commonly associated with
minimalism, particularly with respect to Glass's music.
12
Rebecca Marie Doran Eaton. "Unheard Minimalisms: The Functions of the Minimalist Technique in
Film Scores." PhD diss., University of Texas at Austin, 2008. 102.
13
Eaton, 104.
14
Eaton, 107.
15
Alex Ross. "Sound and Vision; Musical Events," The New Yorker, June 27, 2005.
Nineteen years later, Stephen Daldry's film The Hours is scored by Glass, earning
an Academy Award nomination for best score. The film, based off Michael
Cunningham's Pulitzer prize-winning novel of the same name, weaves the stories of
Virginia Woolf in 1921 (while writing her novel Mrs. Dalloway), a 1950s housewife
(reading Mrs. Dalloway), and a self-effacing woman in 2001 (who lives much like Mrs.
Dalloway herself), dealing with issues of mental and physical disease and suicide. The
opening scene highlights a major theme of the film, the commonality of the characters'
adversities. As the scene jumps from 1921 to 1950 to 2001 in an expository manner,
visually establishing each character as separate but intertwined as a result of the verbal
cues given by Woolf's alternatingly diegetic and non-diegetic voice-overs, Glass's score
paces in the background, aurally uniting the visually disparate scenes.
Again, Eaton provides insightful commentary: "the musical approach of
minimalism is effective at unification because, instead of including recurring melodic
themes, the minimalism itself is heard as a 'theme.' The repetitive, interlocking rhythmic
cells – even if they are different – all appear / sound to the listener to come from the same
source. Lacking melodic leitmotifs, the idea of repeating rhythmic pulsations takes their
place." 16 This can be seen as a thematic parallel to Koyaanisqatsi, where the unification
is of man and machine, rather than temporal gaps. In her excellent article entitled
"Minima Romantica," Susan McClary reinforces this point, referring to the epigraph of
Cunningham's novel, where he in turn quotes Virginia Woolf's diary: "The idea is that the
caves [of character depth] shall connect, & each comes to daylight at the present
16
Eaton, 40.
moment." 17 McClary posits that "it is Glass's music that gives the caves behind the
characters affective depth and that also connects them...," 18 and points out additionally
that Glass self-admittedly "saw his job as lending cohesion to the film's three tangentially
connected stories," in an interview on the film's DVD. 19
The largest departure from Reggio to Daldry is the inclusion of emotion –
whereas Koyaanisqatsi focuses largely on the lack of human emotion shared between
people, The Hours is concerned with exactly the opposite, and this is reflected in the
score by the orchestration. The use of solo piano in a minor key over quiet orchestral
accompaniment is melancholic and romantic in comparison to the mechanical angularity
of the 1983 score. Questions that arise from Koyaanisqatsi, particularly with respect to
how to "feel" about the score, are more easily answered. Arved Ashby, the editor of the
essay collection The Pleasure of Modernist Music in which his own essay appears, asks,
about Koyaanisqatsi, "where is the music going? when will it end? where are the
groupings and the hierarchies? is this passage joyful or sad? is it interpreting what's on
screen, offering a counterpoint to it, or ignoring the visual aspect altogether?" 20 but many
of these questions are answered in the score to The Hours, showing that the spectrum of
minimalism can afford emotions in addition to the hypnotic mechanicism of earlier works
while retaining its unifying abilities, though more powerfully when reflective of the film.
Susan McClary complicates this idea, asserting that in our modern age, "the
ubiquity of minimalism in the soundtracks of our present lives [...] suggests that we have
17
Susan McClary. "Minima Romantica," in Beyond the Soundtrack: Representing Music in Cinema, ed.
Daniel Goldmark et al. (Berkeley: University of California Press, 2007), 57.
18
McClary, 57.
19
Interview with Philip Glass, The Hours, directed by Stephen Daldry (2002; Hollywood, CA: Paramount
Pictures, 2003), DVD.
20
Arved Ashby. "Modernism Goes to the Movies," in The Pleasure of Modernist Music, ed. Ashby. (New
York: University of Rochester Press, 2004), 367.
come to accept such musical processes as natural; indeed, we may not even hear them as
meaning anything at all, just as fish fail to notice the water that surrounds them," 21 but
tempers this by arguing that a cultural practice that denies its own significance is still
meaningful. She eventually argues that the minimalist score with its romantic inflections
serves as a representation of explorable emotional ambiguity, one of the main themes of
the film. 22 Minimalism, while not explicitly representative, can carry meaning through
association. Through this complex lens of film representations of minimalism,
specifically its characteristic ability to indiscriminately associate disparate ideas, its
rejection of individuality, and occasionally emotionless mechanicism that will provide a
standard to evaluate the "minimalism" of Bernard Hermann's scoring techniques.
The claim that repetitious ostinati in films like North By Northwest approaches
minimalism is plausible, and has been made by many. Jack Sullivan, in his exhaustive
examination of every Hitchcock film score, argues that the "manic energy of Herrmann's
score [is] accomplished [...] with a daring minimalism far ahead of its time. Critics didn't
get it: 'The principal motif is repeated ad infinitum,' wrote one, 'and the listener is saved
from acute boredom only by the ever-changing orchestral colors.'" 23 Sullivan points to
the famous crop duster scene as a precursor to "the Philip Glass school," with its wide
open spaces and complete silence, save the wind, cars, and airplane. 24 A History of Film
Music is perhaps the most extensive overview of film music available, and its author
Mervyn Cooke alludes to this idea, mentioning that "In embryonic form, minimalist
techniques are to be found in Herrmann's ostinato-based music in the 1950s [...],
21
McClary, 51.
McClary, 62.
23
Jack Sullivan. Hitchcock's Music (New Haven: Yale University Press, 2006), 236.
24
Sullivan, 240.
22
somewhat prophetic of the style of Philip Glass." 25 Since so many have considered the
idea, the music itself obviously bears some resemblance to the minimalism of Glass and
others, and a cursory viewing of even the overture to North By Northwest is enough to
hear why. The repetitious, rhythmic, and largely tonal structure (though with dissonances
minimalists would avoid) would be easy to pigeonhole as being "minimalistic" in nature,
even if it doesn't sound exactly like the minimalism of the Glass school. When
considering the intention of the overture, the imagery that is presented with it, and the
eventual purpose of the music in the film itself, however, the argument that Herrmann's
scoring is far more modernistic than minimalist becomes far more justifiable than the
contrary.
The solid background of the abstract geometric title sequence by Sam Bass
elegantly fades out during the overture to reveal a shot of a skyscraper. The architectural
lines of the skyscraper has (unbeknownst to the viewer) provided the visual perspective
for the motion graphics, but also can be seen as a skeleton for the overture itself. The
skyscraper, a primary symbol of urbanity, and the geometricity of the title scheme could
be viewed in the same light as Koyaanisqatsi, but this viewpoint is not supported by the
pan down to the main character who, unlike the humans that populate Reggio's world, is
fiercely independent and urbane. In fact, in the same pages that Jack Sullivan wrote of
Herrmann's minimalist trappings, he posits that the overture can be heard as an urban
dance with the hemiola rhythms as a modernist "fandango," transforming the skyscraper
from a representation of monotonous office work to a symbol of "the crazy dance about
to take place between Cary Grant and the world," Hitchcock's own words with regard to
25
Mervyn Cooke. A History of Film Music (Cambridge: Cambridge University Press, 2008), 479.
the title sequence. 26 In a film that is entirely dance and chase, the "minimalistic" ostinati
are revealed to be specifically representative of the clandestine "dance" the film's
characters are participating in.
Whereas a major signifier of film score minimalism is its ability to provide a
blank slate for thematic content, Herrmann is fully representational in his music and, at
the height of his abilities, enhances the movie by doing so. Sullivan goes on to point out
that North By Northwest is one of Herrmann's only films to receive a full love theme and
a "kaleidoscopic" (Herrmann's own word) treatment in terms of orchestration throughout,
giving it a decidedly modernist flavor. 27 Additionally, Sullivan's claim (mentioned
earlier) relating the crop duster scene to minimalism has little basis, and Sullivan himself
provides a better argument that points to Hitchcock's skill as a director and Herrmann's
restraint as a composer: the lack of music in the scene is simultaneously representative of
the lack of modernity in the corn fields and the stark emptiness of the scene itself, a far
more modernist take than minimalist.
This characterization is useful since, as Cooke points out, "Herrmann's formidable
armoury of compositional techniques and freshness of style came more from his classical
background and wide knowledge of twentieth-century concert music;" in other words, he
was a modernist. 28 This is reinforced by the fact that Herrmann, "Like Korngold, did not
compromise the idiom of his concert music when working for films." 29 Perhaps most
damningly, Eaton provides a succinct denial of Herrmann as a minimalist in a footnote:
"This term [minimalism] requires disambiguation; he is referred to this way because he
26
Sullivan, 235.
Sullivan, 236.
28
Cooke, 203.
29
Cooke, 203.
27
used music sparingly, not because his music qualifies as minimalism as a style or
technique. He is known for his ostinatos and for repeating short phrases, but his phrases
are often quite chromatic and are repeated at different pitch levels." 30 Clearly, the
musical rationale for understanding Herrmann's musical language as modernist (if sparse)
greatly outweighs that of minimalism, and the textual or interpretive evidence will
become most obvious in analysis of his masterpiece, Vertigo.
From the opening bars of the overture to the final cadence, the musical and textual
themes are obsession and longing. The shallowest possible listening might categorize the
music as minimalist, if one only listened to the soundtrack, but doing so would be an
insult to the dialectical purpose of the music in the film itself – reducing the repetition
and ostinati to a neat label defeats the purpose of its inclusion. Sullivan, in
acknowledging the "triplets spiraling in contrary motion" and "Herrmann's endless
circlings, recirclings, and suspensions," points out that though "Herrmann's signature
scraps and fragments do appear," the "melancholy elegance of the love music [...] is even
more gripping and obsessive." 31 This is a score that utilizes every "minimalistic" element
in North By Northwest, but also makes it extremely clear that any repetition or silence is
entirely due to Herrmann and Hitchcock's judiciousness as a composer-director pair,
where modernism and romanticism are crashed together to create the torment and longing
the audience experiences along with Vertigo's Proustian main character.
Throughout the film, Herrmann utilizes a number of leitmotifs, and the primary
obsession in the film (the main character's lover, Madeleine) is associated with a leitmotif
30
31
Eaton, 29.
Sullivan, 222.
repeated so frequently, Sullivan refers to it as a ideé fixe a la Berlioz. 32 An ideé fixe could
be interpreted as a minimalistic "seed," but in the context of the film, it is far more
appropriate as a reflection of the main character's ordeal, evidenced by these extramusical
references to Berlioz and Wagner, and musically by distinctly romantic, or even nearly
expressionist sounds. Additionally, Cooke mentions that Vertigo's "fragmentary repeating
patterns are [...] formed into kaleidoscopic musical textures that tread a precarious middle
ground between stability and instability," a clear reference to the film's namesake. When
considering these strong contextual and musical clues that point toward an understanding
of the score in terms of modernism and lyricism in the romantic tradition, the argument
for hearing the music as minimalist falls flat. There is no unifying and hypnotic
undulation here as in Glass's music; this obsession is visceral, and darkly representative
of the thematic content of the film, and it is purposefully done. The importance of this
cannot be understated, and is best reflected in Cooke's assertion that "Herrmann felt
music to be the 'communicating link between the screen and the audience, reaching out
and enveloping all into one single experience,' and in exploiting this link with such
constant resourcefulness he showed how the composer, not the director, could sometimes
be a film's true auteur." 33
Thus, the importance of the differentiation between Glassian minimalism and
Herrmannian modernism becomes clear. Glass's tradition eschews explicit representation,
relying on the marriage of sight and sound to convey meaning, whereas Herrmann is very
nearly a Wagner of the silver screen – the orchestra provides representational and
thematic commentary, heightening the interaction between what the audience sees and
32
33
Sullivan, 223.
Cooke, 212.
hears. Though it is easy to recognize the repetition of musical cells in Herrmann's scores,
the intention, musical execution, and thematic underpinnings of his scores are distinctly
modernist, not minimalist. Glass's music can associate disparate ideas and emphasize the
mechanicism of humanity through relentless repetition, but Herrmann's "minimalism" is a
result of his modernist language capturing the thematic ideas presented in Hitchcock's
films. When considering the musical and contextual arguments, Herrmann's efforts to
bridge the emotional gap between screen and audience cannot be considered minimalistic
in anything but passing resemblance.
As a final thought, it is interesting to consider the conceptual ramifications of
Glass's minimalism and Herrmann's modernism in terms of what led them to write in
each idiom. It seems irreconcilable that, by writing minimalist music, Philip Glass is
representative of an extremely modernist perspective on music, and conversely, that the
modernism exhibited by Herrmann's scores is embroiled in romanticism. This interesting
conflagration of old and new is best viewed through the lens of postmodernist
experimental film in comparison to the golden age of Hitchockian suspense. It is
precisely the alienation of modernism that allowed films like Koyaanisqatsi to be
conceptualized in the first place. In many ways, the simplification of film to the idea of
image and sound, excising any visceral emotional devices, is distinctly minimalist, but
could only be accomplished by a modernist. In contrast, suspense and plot can be seen as
literary tropes of the most traditional sense, where romanticism pushed the boundaries of
emotion and drama. It seems only natural, then, that the extension of the romantic
language first into expressionist screechings and then to modernist hyper-awareness and
formalism would result in film scores exactly like how Herrmann's sound. When
considering this perspective on the perplexing interrelation of modernism as a minimalist
precursor and romanticism's presence in modernism, it becomes clear that the sounds in
the scores of both Glass and Herrmann can be completely anticipated as a result of their
perspectives on humanity as it relates to the art of filmmaking.
Works Cited
Ashby, Arved. "Modernism Goes to the Movies," in The Pleasure of Modernist Music,
ed. Ashby. New York: University of Rochester Press, 2004.
Cooke, Mervyn. A History of Film Music. Cambridge: Cambridge University Press, 2008.
Eaton, Rebecca Marie Doran. "Unheard Minimalisms: The Functions of the Minimalist
Technique in Film Scores." PhD diss., University of Texas at Austin, 2008. 102.
Glass, Philip. "Interview." The Hours, directed by Stephen Daldry. Hollywood, CA:
Paramount Pictures, 2003. DVD.
McClary, Susan. "Minima Romantica," in Beyond the Soundtrack: Representing Music in
Cinema, ed. Daniel Goldmark et al. Berkeley: University of California Press, 2007.
Reggio, Godfrey. Dir. Koyaanisqatsi. 1983; Santa Monica: MGM Home Entertainment,
2002. DVD.
Ross, Alex. "Sound and Vision; Musical Events," The New Yorker, June 27, 2005.
Sullivan, Jack. Hitchcock's Music. New Haven: Yale University Press, 2006.
Links to Senior Recital:
https://www.youtube.com/watch?v=OpWa4PiHe-Q
https://www.youtube.com/watch?v=V3WN41v5IZk
https://www.youtube.com/watch?v=K8j0-f3jshM
https://www.youtube.com/watch?v=VOaaI-UZGHg
MATH420 Project – Image Compression
Aidan Meacham
cba – April 14, 2014
1
Introduction
Digital raster images are 2D matrices of values that, when printed, compose the image we
are familiar with. When these arrays become large, either in resolution or, in the case of
color images, by having multiple matrices implicit in a single image, it becomes useful to
“compress” images. This is accomplished by representing them in a smarter way than the
characteristically dense manner the original matrix is captured. Methods such as SVD (and,
in turn, Principal Component Analysis) as well as Discrete Cosine Transforms (DCT, with
an emphasis on quantization) can give greatly compressed images with little loss of data.
Through the use of linear algebra, these processes can be applied through matrix operations,
with varying degrees of speed and stability. This project will explore the mechanics of these
methodologies with analysis of the linear methods used to apply them quickly and stably.
It is becoming increasingly important in the modern era to be able to compress data,
even with expanding storage solutions and increased processing power available. As digital
imaging devices’ sensors increase in size, the demands for storage space increase exponentially. The storage and manipulation of these files therefore benefits greatly from methods
to reliably and speedily compress (and decompress for display) extremely large files.
2
Preliminaries
Before compressing images, it is important to know the form the images take, which we will
use to examine them throughout this project. For example, a simple definition of a black and
white image is as follows: an image of resolution m × n is represented as a matrix of values
(often the range of a byte, 0 to 255, which has a convenient hexadecimal representation
frequently used in image editing) where the number is an intensity value corresponding to a
particular grayscale shade. For a color image, depending on the color space used, consider
separate intensity matrices for each color. For example, in the RGB color space (as in a
1
computer screen), a single image has a matrix for the intensity of each color channel that
corresponds to the brightness of a red, blue, and green pixel. (Imagine “00”(hex) as a pixel
with no brightness and “FF” as maximum brightness, so “FFFFFF” would display a white
pixel and “FF9900” would be a bright orange.) Compression is the manipulation of these
values into convenient representations, frequently with a great number of small or zero values.
There are two general categories of compression available, lossy and lossless, whose names
are indicative of whether or not in the compression process any information is discarded and
unrecoverable. Typically, lossy algorithms sacrifice quality for file size, whereas lossless algorithms typically cannot achieve the same high rate of compression, but maintain bit-perfect
recovery of an original, uncompressed file. Particularly for images, the specific contents of
a file may be suited to one algorithm or another, making it difficult to tout one algorithm
as the “best” for any given situation. Therefore, a variety of methodologies and approaches,
including more generalized digital file compression techniques that are not discussed in great
detail here, are both necessary and convenient.
3
SVD
The first methodology which can be used in the compression of images is the SVD, or singular
value decomposition. While relatively expensive computationally, as it requires the computation of eigenvalues, this method provides stable compression and a useful demonstration
of linear algebra as a compression tool. One of the immediate advantages of the SVD is
that, unlike many matrix decompositions, it can be found for any matrix regardless of size
or singularity. This relies on the fact that the matrix-adjoint products are always positive
semi-definite, giving positive eigenvalues, and thus, the ability to reliably return singular
√
values, λ. A definition of the decomposition follows, with a discussion of its assumptions
and implications.
√ √
√
Definition 1 (SVD). A is a matrix with singular values σ1 , σ2 , . . . , σr , where r is the
rank of A∗ A. Define V = [x1 |x2 | . . . |xn ], U = [y1 |y2 | . . . |yn ] where {xi } is an orthonormal
√
basis of eigenvectors for A∗ A and yi = √1σi Axi . Additionally, si = σi .
2








S=







s1
0
s2
...
sr
0
0
..
.
0
0







.






Thus, AV = U S
A = U SV ∗ .
The general idea of the approach of the SVD is that a given matrix can be constructed as
the product of a rotation, scaling, and a second rotation of the identity matrix. The axes that
are scaled to a greater degree correspond to greater singular values, and by ordering them
from greatest to least, one can see the most significant “contributions” to the makeup of a
matrix from the given orthonormal bases. This can be seen most readily in the summation
formation it can take, which we will see next.
For image compression purposes, the most useful aspect of the SVD is its “truncated
r
P
form,” a specialization of this expression of the product derived above: A =
si xi y∗i ,
i=1
where r is the rank of A∗ A and the si are ordered in decreasing magnitude, s1 ≥ s2 ≥ · · · ≥
sr . For i < r, this neglects the lower weighted singular values, and provides a very good
approximation of an image, a comparison of which can be found below, despite its extremely
lossy nature. This is sometimes referred to as a version of the Karhunen-Loeve transform,
which we will see again with regard to principal component analysis and the modal and
covariance matrices. Once the desired number of singular values has been decided upon, the
unnecessary singular values and the corresponding columns of U and V can be discarded,
decreasing the amount of storage necessary to reconstruct the image.
4
SVD Example
To finish our discussion of the SVD, sample calculations detailing the process of compression via the decomposition in Sage are given below, with the resulting images provided for
comparison. We will work in Sage, utilizing the pylab package. First, import the image and
convert it to a Sage matrix in order to perform operations.
import pylab
3
A = pylab . mean ( pylab . imread (DATA + ’ cameraman . png ’ ) , 2 )
B=matrix (A)
Next, perform the singular value decomposition and reconstruct the image from the number
of desired singular values.
u , s , v = B .SVD( )
n = 32 ‘ ‘ ‘ ‘ ‘ ‘ Number o f SVs used ”””
C = ra n g e ( n )
f o r j i n range ( n ) :
C[ j ] = ( ( u [ : , j ] ∗ v . t r a n s p o s e ( ) [ j , : ] ) ∗ s [ j , j ] )
D=sum (C)
m a t r i x p l o t (D)
The original image (of dimension 256 × 256) has 256 components, so the first image here
is an exact reconstruction of the original, then descending in both number of elements and
quality.
Cameraman, 256 elements
Cameraman, 128 elements
Cameraman, 64 elements
Cameraman, 32 elements
4
Cameraman, 16 elements
5
Cameraman, 8 elements
Principal Component Analysis
Principal Component Analysis can be seen as a method similar to SVD compression in the
sense that it selects the “most important” components, however, the method for doing so is
different. PCA can be seen as a statistical process for finding the best representation for a
set of data. Principal components have a wide variety of applications in many fields, such as
finding the principal moments and axes of inertia in physics. Generally, however, the process
of finding the principal components of a set of data encompasses the same basic idea, and
typically involves the solution of an eigenvalue problem.
Typically, the goal of PCA is to find an orthonormal basis for a space which orders
variables in decreasing order of their variance. In terms of information theory, the idea of a
variable’s entropy (conceptually introduced by Claude Shannon in 1948) is the basis for PCA,
wherein variables that have greater variance (or, higher entropy) carry more information,
and therefore, maximizing the variance of a particular variable will maximize the density
of information it can carry. By ordering variables in order of decreasing variance, we can
compress data via an approximation by leaving off the components that contribute the least
information, which are exactly those with low variance. In this manner, PCA is “just like”
the SVD, which places precedence on singular values of higher weights.
One method for finding the aforementioned components is through the modal matrix,
which will follow as a result from a short foray into some statistics. Since we are primarily
working with matrices m×n with discrete values, we will use the discrete case. The expected
P
value of a random variable, E(X) =
xi p(xi ) = µ. This value can be viewed as a mean of
sorts, predicting an average outcome for a given probabilistic scenario. Variance, V (X), is
E[(X − µ)2 ], and can be viewed as the expected deviation from the mean, µ. The positive
square root is the familiar standard deviation. Next, the covariance or correlation of two
5
variables is Cov(X, Y ) = E[(X − µx )(Y − µy )], and if this is zero, X and Y are independent.
If X = Y then we recover our earlier definition of variance.
P
The covariance matrix of X is Cov(X) or
= E[(X − µ)T (X − µ)] where µ is the
vector of expected values µi = E(Xi ). This matrix is positive semi-definite, which means its
eigenvalues will also be positive, and similar to the SVD, these are what we will order. This
is where the SVD comes into the calculations for PCA and why the K-L transform can be
seen as both SVD and PCA compression. Additionally, Cov(X) is symmetric, and therefore,
diagonalizable. The eigenvectors of the covariance matrix of X must be orthogonal, and
by scaling can be made into an orthonormal set. Setting these scaled eigenvectors as the
rows of a matrix creates the modal matrix M , whose rows are the principal axes for X, and
diagonalize the covariance matrix Cov(X).
Theorem 1. PCA Finds Principal Axes (After Hoggar [1])
Let the orthonormal eigenvectors of Cov(X), where X = X1 , . . . , Xd , be R1 , . . . , Rd .
Let X have components (in the sense of projection) {Yi }, where Y = Yi .
Then {Ri } is a set of principal axes for X.
Proof.
Yi = X · Ri = XRiT
Y = XM T , M = Rows(Ri ).
Because M diagonalizes Cov(X), we can write:
Cov(Y ) = Cov(XM T ) = M Cov(X)M T ,
which is a diagonal matrix of eigenvalues. Additionally, this shows that V (Yi ) = λi .
If the Ri are the principal axes for X, then the Yi will be the principal components, and we
can expect them to be uncorrelated, meaning the variance of X · Ri is maximal. This is only
true when, for an arbitrary R, R = Ri , meaning that they are the principal axes for X.
Additionally, if E(X) 6= 0, one can subtract E(X) from X, perform these calculations,
and add E(X) back. If we have d vectors X, we can transform them into k vectors Y , k < d
by discarding the Yk+1 to Yd vectors with a minimal loss of data. This is similar to SVD
compression, but instead of using the eigenvectors of a matrix itself as a basis (in combination
with singular values), we eliminate the singular values for the statistically chosen basis of
eigenvectors. Essentially, this process, which we can now formally call the K-L transform,
in the words of Hoggar, “minimizes the mean squared error for mapping d-vectors in a given
class into a space of dimension k” [1, 297].
6
Often in image compression, blocks of 8 × 8 pixels are selected and turned into vectors
of length 82 = 64. These N vectors are stacked as rows into a “class matrix” HN ×64 after
subtracting the mean, then the modal matrix M is calculated, either by the method described
before or through the SVD. (If the dimension of these vectors is greater than N , performing
the same calculations with HH ∗ is a quicker computation.) Once we have acquired the
principal components, we can project our data using as few or as many principal components
as we like via matrix multiplication. Similar to the discarding of unnecessary components
in SVD compression, the reduction of vector space dimension allows the image to be stored
much smaller than the original before being reconstituted.
Due to the similarity to SVD compression, we will not examine a method to perform the
Karhunen-Loeve transform in detail here, however, many authors have provided algorithms
to do so, including S. Hoggar [1] and Mark Richardson [7], including illustrations of comparable quality to the SVD. One caveat, as pointed out by Hoggar, is that performing the K-L
transform utilizing SVD rather than diagonalization is numerically stabler, at the cost of a
lengthier initial computation, than other methods, such as the Discrete Cosine Transform,
which we will detail next.
6
Discrete Cosine Transform
The final method of image compression we will examine is perhaps the most popular in
practical usage as it is utilized by the JPEG file format. The discrete cosine transform is of
the family of fast Fourier transforms, and like the other transformations we have examined,
behaves linearly, allowing us to write a matrix form that is quick to compute. The onedimensional DCT can be written as follows, where φk is a vector with components n, written
as a variable to avoid confusion with matrix notation.
 q
2

cos (2n+1)kπ
, for n = 1, 2, . . . , N − 1,
N
q 2N
φk (n) =
1

,
for n = 0.
N
From this definition, a set of k vectors (each of dimension n) is orthonormal and spans
the space of N -vectors. Because of this, we can easily invert the matrix of columns M =
[φ0 |φ1 | . . . |φN −1 ] by its transpose. The matrix M is applied via the matrix-vector product,
transforming input vectors which can easily be reverted through a matrix-vector product
with the inverse of M . This allows us to extend the DCT to a 2D case, where a matrix of
values can be transformed via the calculation B = M AM T .
Since the 2D case of DCT is simply a composition of the same function along each
dimension, the product is separable, and is therefore comparable to applying the 1D case
7
twice. In the case of an image, we are essentially performing the same operation on the
rows, then columns, but through matrix multiplication instead of repeated matrix-vector
products. One method for image compression utilizes a similar formulation by partitioning
an image into vectors of length 8, first by rows, then columns, and applying an 8 × 8 DCT
matrix to each vector. By the end, each 8 × 8 submatrix of the image has been transformed,
accomplishing the same ends as the matrix version we will explore here.
The JPEG file format utilizes the aforementioned method for the compression of images.
The primary transformation applying the DCT achieves is in moving information to the
earliest indices of a vector or matrix, leaving many of the latter entries close to zero. The
lossy part of JPEG compression happens when many of these “close to zero” entries are set
to zero, depending on the level of compression desired, a step called quantization. In the
case of the twice-applied 1D version, a particular compression setting would force the last n
indices of a vector to zero, meaning out of every 8 × 8 submatrix, only (8 − n)2 coefficients
out of 64 would be nonzero.
Figure 1. Zigzag Ordering
At this point, the JPEG format continues to compress the data via zigzag reordering
of the coefficients, and then applying (typically) Huffman encoding to the resulting array.
Zigzag reordering (pictured above) is used before encoding to take advantage of the plethora
of zeroes in the larger indices of the transformed and quantized matrix. By storing coefficients
in the order taken by the zigzag path, the zeros are generally concatenated at the end of
the list, which is helpful for the next step of the process. From here, Huffman encoding
(which is a lossless entropy-based algorithm) further reduces the storage space needed for
the reorganized arrays, organizing the reordered data through a variable-length code table.
The only step of this process that is not reversible is the quantization of the transformed
vectors, but reversing every other step recovers the image. This means that even if one kept
all 64 coefficients, there would be some loss of quality, as information is being discarded
by rounding in any case. In this manner, an image can be compressed a great deal, but
because the way the DCT transforms frequencies rather than intensities (which the human
eye is much more sensitive to), the recovered image retains a high degree of recognizability.
8
The most noticeable downside of JPEG compression are the “blockiness” of the 8 × 8 pixel
arrays. If the compression level is sufficiently high, the blocks begin to look averaged across
the entirety of their values, losing a degree of smoothness in areas of little change and leading
to characteristically blocky looking artifacts. For natural images with a high information
density, this is not especially apparent, but for a manufactured image (such as text) that has
a high degree of contrast, the block algorithm JPEG uses is not well suited for high-quality
image recovery.
7
DCT Example
To finish, we will apply the DCT as an example, with some code provided. As before, we
utilize pylab to import the image.
import pylab
A = pylab . mean ( pylab . imread (DATA + ‘ k l e i n r e s i z e d . png ’ ) , 2 )
B = matrix (A)
Next, create the 8 × 8 DCT matrix, M .
M=matrix (RR, 8 )
from s a g e . s y m b o l i c . c o n s t a n t s import p i
PI = p i
f o r i i n range ( 1 , 8 ) :
f o r j i n range ( 8 ) :
M[ i , j ] = ( s q r t ( 1 / 4 ) ∗ c o s ( ( 2 ∗ j +1)∗ i ∗PI / ( 2 ∗ 8 ) ) )
f o r j i n range ( 8 ) :
M[ 0 , j ] = 1/ s q r t ( 8 )
Now we will partition our image into 8 × 8 blocks. Since our image has dimension 200 × 160,
there will be 20 blocks along the horizontal and 25 along the vertical for a total of 500 blocks.
It will then be multiplied with the DCT matrix, the product of which is shown here.
y = r an ge ( 5 0 0 )
f o r j i n range ( 2 0 ) :
f o r i i n range ( 2 5 ) :
y [ j +20∗ i ]=B [ ( i ∗ 8 ) : ( i ∗8+8) ,( j ∗ 8 ) : ( j ∗8+8)]
x = r an ge ( 5 0 0 )
f o r i i n range ( 5 0 0 ) :
x [ i ]=M∗y [ i ] ∗M. t r a n s p o s e ( )
DCT = b l o c k m a t r i x ( 2 5 , 2 0 , x )
9
m a t r i x p l o t (DCT)
One can easily see the “compression” of data to the lowest indices of the 8 × 8 blocks. From
here, the image is quantized (shown below is the case where 5 elements remain; retaining
all 8 would reproduce the original image seen below). JPEG compression would reorder the
block coefficients and compress the resulting data, but we will skip this to show the final
image below, with increasing quantization levels, reconstructed via the inverse DCT.
x2 = r ange ( 5 0 0 )
f o r i i n range ( 5 0 0 ) :
x2 [ i ]=x [ i ]
f o r i i n range ( 5 0 0 ) :
x2 [ i ] [ : , 5 : 7 ] = 0
f o r i i n range ( 5 0 0 ) :
x2 [ i ] [ 5 : 7 , : ] = 0
x3 = r ange ( 5 0 0 )
f o r i i n range ( 5 0 0 ) :
x3 [ i ] = M. t r a n s p o s e ( ) ∗ x2 [ i ] ∗M
image = b l o c k m a t r i x ( 2 5 , 2 0 , x3 )
m a t r i x p l o t ( image )
The image is very good until the last, where the 8 × 8 blocks are especially apparent; at this
point, they are the average value of the entire block.
10
8
Klein, all 8 elements
Klein, 5 elements
Klein, 3 elements
Klein, 1 element
Notes & Miscellany
A complete analysis of the techniques above and the general state of image compression
(at least for techniques that could be considered linear) would require some discussion of
encoding, particularly Huffman encoding and LZW compression. We will not go into the
details here, however, note that some of the techniques described in this paper in actual
application are designed to take full advantage of these additional steps, and may not be
identical to the simplified academic versions discussed above. For example, the GIF file
format relies heavily on LZW compression, which is essentially a dictionary lookup algorithm
11
that works surprisingly well for natural images. The images used for the example calculations
in this paper are in the PNG format, a spiritual successor to the GIF file format.
9
Conclusion
Through our discussion of two similarly styled compression methods (SVD and PCA) that
make use of matrices’ natural characteristics (eigenvalues), as well as an industry standard
transform (DCT for JPEG), we have seen a variety of approaches to the multifaceted subject
of compression, at least for natural images. As is apparent from the nature of the calculations
being performed, linear algebra is particularly well-suited to the task, and the methods
developed take full advantage of the speed and accuracy afforded by linear operations. A
logical next step in a fuller understanding of compression could focus on wavelets and fractals,
as well as entropy compression and information theory. However, for a basic understanding of
the principles and methods of image compression, the three methods discussed here represent
a useful and straightforward introduction to the art.
10
References
[1] Hoggar, S. G. Mathematics of digital images: creation, compression, restoration, recognition. Cambridge: Cambridge University Press, 2006.
[2] N. Ahmed, T. Natarajan and K.R. Rao. “Discrete Cosine Transform.” IEEE Trans.
Computers, 90-93, Jan. 1974.
[3] Lay, David. Linear Algebra and its Applications. New York: Addison-Wesley, 2000.
[4] Trefethen, Lloyd N., and David Bau. Numerical linear algebra. Philadelphia: Society
for Industrial and Applied Mathematics, 1999.
Additional web resources:
[5] http://introcs.cs.princeton.edu/java/95linear/
[6] http://www.uwlax.edu/faculty/will/svd/
[7] http://people.maths.ox.ac.uk/richardsonm/SignalProcPCA.pdf
11
License
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International
License. To view a copy of this license, visit http://creativecommons.org/licenses/
by-sa/4.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain
View, California, 94041, USA.
12
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Linear Methods for Image Compression
Math 420, Prof. Beezer
Aidan Meacham
University of Puget Sound
Methods
SVD
PCA
DCT
Outline
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Methods
SVD
PCA
DCT
Outline
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Methods
SVD
PCA
DCT
RGB Color Space
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
I
Intensity and Representation
I
Gamut mapping and Translation
I
Absolute Color Spaces
Lossy vs. Lossless Methods
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
I
Lossless Methods - GIF / LZW
I
Usefulness of Lossy Compression
I
Limit - arithmetic, entropy, and LZW coding
Outline
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Methods
SVD
PCA
DCT
Linear Methods for
Image
Compression
SVD
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
Definition
I
I
√
√ √
A is a matrix with singular values σ1 , σ2 , . . . , σr ,
where r is the rank of A∗ A and σi are eigenvalues of A
Define V = [x1 |x2 | . . . |xn ], U = [y1 |y2 | . . . |yn ] where
{xi } is an orthonormal basis of eigenvectors for A∗ A
and yi = √1σi Axi
SVD
PCA
DCT
Linear Methods for
Image
Compression
SVD
I
Additionally, si =
√
Aidan Meacham
σi
Preliminaries
I
Color Spaces
Lossy vs. Lossless







S =







s1
0
s2
..
.
sr
0
0
..
.
0|
I
Thus,
AV = US
A = USV ∗
0













Methods
SVD
PCA
DCT
SVD Truncated Form
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
I
A=
r
P
i=1
Methods
si xi y∗i , where r is the rank of A∗ A and the si
are ordered in decreasing magnitude, s1 ≥ s2 ≥ · · · ≥ sr
I
For i < r , this neglects the lower weighted singular
values
I
Discarding unnecessary singular values and the
corresponding columns of U and V decreases the
amount of storage necessary to reconstruct the image
SVD
PCA
DCT
SVD Example
Sage
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
I
Import image and convert to Sage matrix
I
Perform SVD decomposition
I
Choose number of singular values and reconstruct
Linear Methods for
Image
Compression
SVD Results
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Cameraman, 256 elements
Cameraman, 128 elements
Linear Methods for
Image
Compression
SVD Results
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Cameraman, 64 elements
Cameraman, 32 elements
Linear Methods for
Image
Compression
SVD Results
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Cameraman, 16 elements
Cameraman, 8 elements
Principal Component Analysis
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
I
Wide variety of applications in many fields:
I
I
I
I
Principal moments and axes of inertia in physics
Karhunen-Loeve Transform in signal processing
Predictive analytics - customer behavior
Statistical method for maximizing “variance” of a
variable; similar to SVD
SVD
PCA
DCT
Statistics / Information Theory
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
I
Variables with greater variance (higher entropy) carry
more information
I
Maximizing variance maximizes the information density
carried by one variable
I
Compress data via approximation, leaving off less
significant components
I
Weighting - similar to SVD
SVD
PCA
DCT
Statistics / Information Theory
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
P
I
E (X ) =
xi p(xi ) = µ
I
“Mean;” average outcome for a given scenario
I
V (X ) = E [(X − µ)2 ]
I
Expected deviation from the mean, µ
I
Positive square root is standard deviation
SVD
PCA
DCT
Statistics / Information Theory
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
P
= E [(X − µ)T (X − µ)]
I
Cov(X ) or
I
µ is the vector of expected values µi = E (Xi )
I
This matrix is positive semi-definite, which means its
eigenvalues will also be positive
I
Cov(X ) is symmetric, therefore, diagonalizable
I
Modal matrix M, composed of rows of eigenvectors for
Cov(X ), diagonalizes the covariance matrix
SVD
PCA
DCT
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Theorem (PCA Finds Principal Axes, via Hoggar[1] )
I
Let the orthonormal eigenvectors of Cov(X ), where
X = X1 , . . . , Xd , be R1 , . . . , Rd
I
Let X have components (in the sense of projection)
{Yi }, where Y = Yi
I
Then {Ri } is a set of principal axes for X
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Linear Methods for
Image
Compression
Proof.
Aidan Meacham
Preliminaries
Yi = X · Ri =
XRiT
Y = XM T , M = Rows(Ri ).
Because M diagonalizes Cov(X ), we can write:
Cov(Y ) = Cov(XM T ) = MCov(X )M T ,
which is a diagonal matrix of eigenvalues; V (Yi ) = λi .
If the Ri are the principal axes for X , then the Yi will be the
uncorrelated principal components, meaning the variance of
X · Ri is maximal.
For an arbitrary R, this is only true whenR = Ri , so {Ri } are
the principal axes for X .
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
PCA Compression
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
I
Given d vectors X , transform into k vectors Y , k < d
I
Discard Yk+1 to Yd vectors with a minimal loss of data
I
Blocks of 8 × 8 pixels selected; turned into vectors of
length 82 = 64
I
N vectors stacked as rows into a “class matrix” HN×64
after subtracting the mean
I
Calculate modal matrix, then project data using as
many principal components as we like
Methods
SVD
PCA
DCT
PCA Remarks
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
I
Less stable than SVD
I
Better for extremely large data sets
I
Big data - consumer modeling
Discrete Cosine Transform
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
Definition
The one-dimensional DCT can be written as follows, where
φk is a vector with components n, written as a variable to
avoid confusion with matrix notation
 q
(2n+1)kπ
2

, for n = 1, 2, . . . , N − 1,
N cos
q 2N
φk (n) =
1

,
for n = 0.
N
SVD
PCA
DCT
DCT Properties
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
I
A set of k vectors (each of dimension n) is orthonormal
I
The matrix of columns M = [φ0 |φ1 | . . . |φN−1 ] is
invertible by its transpose
I
2D case: apply transformation first to rows, then to
columns (separable; composition of function along each
dimension)
I
A matrix of values can be transformed via the
calculation B = MAM T
SVD
PCA
DCT
JPEG File Format
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
I
JPEG utilizes DCT
I
Applying DCT moves information to lower indices
(vector or matrix)
I
Higher index entries close to zero
I
Lossy compression - quantization
I
Settings - force the last n indices of a vector to zero
I
For every 8 × 8 submatrix, (8 − n)2 coefficients out of
64 nonzero
Methods
SVD
PCA
DCT
JPEG Continued
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
I
The transformed array undergoes zigzag reordering to
take advantage of zeroes in the larger indices
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
I
I
This array is compressed via Huffman encoding (lossless
entropy-based algorithm)
I
Huffman encoding utilizes a variable-length code table
to construct a frequency-sorted binary tree
JPEG Remarks
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
I
Only non-reversible step is quantization
I
Reversing other steps (switching order of multiplication)
retrieves image
I
Data lost no matter what - rounding errors
I
DCT transforms frequencies, not intensities - human
eye sensitivity / recognition
I
Blocky artifacts - natural vs. manufactured images
SVD
PCA
DCT
DCT Example
Sage
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
I
Import image and convert to Sage matrix
I
Create DCT matrix
I
Subdivide matrix and apply transform
I
Quantize
I
Reconstitute
Linear Methods for
Image
Compression
DCT Applied
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Klein, 8 elements
Klein, post-DCT
Linear Methods for
Image
Compression
DCT Results
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Klein, 8 elements
Klein, 5 elements
Linear Methods for
Image
Compression
DCT Results
Aidan Meacham
Preliminaries
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
Klein, 3 elements
Klein, 1 element
References
Linear Methods for
Image
Compression
Aidan Meacham
Preliminaries
I
[1] Hoggar, S. G. Mathematics of digital images:
creation, compression, restoration, recognition.
Cambridge: Cambridge University Press, 2006.
I
[2] N. Ahmed, T. Natarajan and K.R. Rao. “Discrete
Cosine Transform.” IEEE Trans. Computers, 90-93,
Jan. 1974.
I
[3] Lay, David. Linear Algebra and its Applications.
New York: Addison-Wesley, 2000.
I
[4] Trefethen, Lloyd N., and David Bau. Numerical
linear algebra. Philadelphia: Society for Industrial and
Applied Mathematics, 1999.
Color Spaces
Lossy vs. Lossless
Methods
SVD
PCA
DCT
crosscurrents literary and art magazine
deadline april 3, midnight «» photoshoot april 1, 4pm
ccr@pugetsound.edu
Crosscurrents
puget sound’s literary and arts magazine
email ccr@pugetsound.edu
april 9, 2014
Crosscurrents
puget sound’s literary and arts magazine
up to 4 art + 3 poetry
+ 2 prose + 1 other
email submissions to
ccr@pugetsound.edu
deadline: april 4, 2014
asups cultural events presents
a cuban musical experience
La Familia
valera miranda
October 26, 7:30 ~ Homecoming Weekend
Kilworth Chapel ~ University of Puget Sound
$4 w/ UPS ID, $12 Public
Tickets available at the Puget Sound info center,
online at tickets.pugetsound.edu and at the door
For accessibility information, please contact accessibility@pugetsound.edu or call (253)-879-3236.
aidan
plays
some
cello
sunday
february 24
2 pm schneebs
brahms
beethoven
cassado
aidan
plays
cello
again
sunday march 9
2 pm schneebs
reception to follow
bach
zavortink
janáček
martinů
Aidan Meacham
4215 61st ST CT NW
Gig Harbor, WA 98335
253.381.4472 (c)
reachaidanmeacham@gmail.com
Objective:
As a physicist and musician with a serious interdisciplinary interest in communication through graphic design and programming, I hope to broaden my
experiences while offering my curiosity and energy in today’s rapidly changing
workplace. With a wide variety of skills ranging from web design to culinary preparation to high-definition record engineering, I am well-prepared and excited to
take on creative and challenging tasks and engage fully with coworkers and clients.
Education:
University of Puget Sound, Class of 2014
B.S. in Physics and B.M. in Cello Performance
Relevant Coursework:
Theoretical and experimental (lab-based) physics
Mathematics including advanced linear algebra with
a focus on applied programming
Advanced orchestral conducting, chamber music,
and individual cello studies
Critical thinking and writing-based courses in literature and sciences
Relevant Skill Areas:
Graphic Designer – 4 years of visual communication
experience at Crosscurrents, University of Puget
Sound’s literary and arts magazine
Office & Beyond – proficient in all Microsoft Office
applications and rapid communication
Computers & Programming – practical experience
with python and other object-based languages,
web design (aidanmeacham.com), experienced
computer and networking troubleshooter
Organizational Skills – interested in efficiency
through automation and optimized workflow
Other Experience:
Scientific data modeling, computational physics and
applied linear algebra through Sage
Sound and video recording / live-streaming, cinematography, lighting, and electronics experience
Recent Employment:
Layout Director, Crosscurrents Review (2010 – Current)
Orchestral Recital Series Cellist, Tacoma Music Teachers Association (2012 – current)
Graphic Designer, ASUPS (2013 – Current)
Remote Location Staff, ESS Support Services
Worldwide (2013 – 2014)
Cello Teacher, Peninsula Youth Orchestra (2012 – 2014)
Sailing Instructor, Tacoma Yacht Club (2009 – 2013)
Other Interests:
Acoustics, sound and music recording technology,
high-definition sound reproduction
Classical and contemporary chamber music, 16 years
of pop and rock cello playing
Theater and opera as an actor, designer, and musician
Dinghy sailing, instruction and competition on the
northwest circuit
Data visualization and predictive analysis
Print and web media advertising
Independent, foreign, and classic film
Backyard Projects:
Atypical musical instruments, “circuitbending”
DIY high-speed photography
Stop-motion animation and vfx with LEGO
Molecular gastronomy