Poster

Oblique Random Forests for 3-D Vessel Detection
Using Steerable Filters and Orthogonal Subspace Filtering
Matthias Schneider1, Sven Hirsch1, Gábor Székely1, Bruno Weber2, and Bjoern H. Menze1
1 Computer Vision Laboratory, ETH Zurich, Switzerland
2 Institute of Pharmacology and Toxicology, University of Zurich, Switzerland
COMPUTER AIDED AND IMAGE GUIDED
MEDICAL INTERVENTIONS
Motivation
Vessel detection and segmentation
Objectives
• Crucial for many clinical diagnostic and planning tasks
• Challenges:
- Multiscale nature of blood vessels
- Contrast inhomogeneities
- Image noise and artifacts
• Clinical studies on neurodegenerative diseases (e.g.
Alzheimer’s disease) make use of high-resolution imaging
to analyze microvascular structures of the brain.
► Efficient tools for vessel segmentation and analysis required
- Vessel segmentation framework
- Applicable for 2-D/3-D data
- Efficient computation
- Scalable (high-resolution datasets)
Figure 1: Left: Medical images showing different vascular structures: Fundus photography (top) and fluoroscopy of the left
coronary artery (bottom). Right: Scanning electron micrograph of a vascular corrosion cast from the macaque primary visual
cortex [6]. Arteries are shaded in red and veins are blue. The circular image shows an axial slice of a cylindrical sample punched
out of the cortex (superimposed cylinder) obtained by synchrotron radiation X-ray tomographic microscopy (srXTM) [9].
Method
Machine Learning Framework
Orthogonal Subspace Filters (OSF)
• Learn “optimal” eigenfilters from
local vessel patches
• Orthogonal basis preserving
maximum variance
• Vessel appearance described in
low-dimensional feature space
Steerable Filter Templates (SFT)
• Design parameterized filter templates similar to
highly structured OSF eigenfilters in order to better
model the problem and explain the data [5]
• Parameterization: Gaussian derivatives up to order M
Random Forest Classifier [1]
• Orthogonal splits [1]
- Optimal thresholds on randomly
selected single features
- Orthogonal 1-D hyperplanes
• Advantages:
- Explicit scale parameterization
- Steerability allows for efficient directional filtering
• Oblique splits [7]
- Multidimensional hyperplanes
- Linear regression with elastic-net penalty to learn
multivariate (optimal) split direction [4]
- Better generalization in areas with few samples
Why Random Forests?
• Efficient, scalable, and accurate
• Continuous posterior (”confidence”)
• Very few parameters involved
• Estimates of feature importance and
generalization error during training
Figure 3: Machine learning framework for 3-D
vessel segmentation using OSF and SFT features
along with a random forest (RF) classifier. The
steerable SFT filters can efficiently be applied
along the normalized vessel direction (Rθ,φ).
Figure 2: Visualization of 3-D filter templates. Top: OSF eigenfilters learned from
vessel patches. Bottom: SFT templates at fixed scale up to order two. Templates are
plotted for centered sagittal, coronal, and axial 2-D slices.
Results
σ =1 σ =2 σ =3 σ =4 σ =5 σ =6 σ =7 σ =8
10
10
−1
10−2
10−3
10−4
10−5
1
10
19
28 37 46
Feature index
55
64
k
Gσ1,0,0
σk
G1,1,0
k
Gσ1,1,1
k
Gσ2,0,0
k
Gσ2,1,0
k
Gσ2,1,1
σk
G2,2,0
k
Gσ2,2,1
k
Gσ2,2,2
1
RF-OSF
d=9
d = 27
d = 57
d = 102
RF-SFT
M =1
M =2
M =3
M =4
Frangi
Sato
Otsu
maximum F1
0.8
Precision
0
Importance
Image Data
• Four high-resolution 3-D datasets
• Synchrotron radiation X-ray tomographic microscopy (srXTM)
• Cylindrical samples of the murine
somatosensory cortex
• Dimensions (2048px)3, 700nm isotropic voxel spacing, ROI (256px)3
0.6
0.4
0.2
0
0.5
72
0.6
0.7
0.8
Recall
Figure 4: Visualization of decision boundaries (solid black) for a
two class problem (red/green) using a random forest classifier
along with orthogonal splits (left), oblique splits in a single
decision tree (center) and an ensemble of decision trees (right) [7].
The dashed black line indicates Bayes’ optimal border.
Figure 6: Feature relevance (permutation importance [1]) of the RF-OSF (left) and RF-SFT
model (right) on a logarithmic scale (oblique splits). Right: Precision-recall curves and optimal
operating points w.r.t. F1 measure for RF-OSF and RF-SFT models (oblique splits) with varying
number of features in comparison to (optimized) Frangi’s/Sato’s vesselness [2], and Otsu
thresholding [8].
0.9 0.95 1
Experiments
Orthogonal Splits
SFT Features
• OSF model: Eigenfilters computed from 3000 randomly • Fast training
• Second order Gaussian derivasampled vessel patches of size (19px)3
• Very fast split evaluation
tives highly discriminative
• SFT model: Gaussian derivatives up to order M=1,2,3,4 at • Relatively deep trees, particularly
• Clearly outperforms OSF features
three scales resulting in 9 (27, 57, 102) features
for highly correlated features
w.r.t. segmentation performance
• RF training
- Otsu thresholding used to compute training labels Oblique Splits
- Training set: 4000 randomly sampled FG/BG samples • Split evaluation and finding optimal
- 256 trees
split parameters more expensive
• Ground truth
• Better generalization and accuracy
- Semi-automatic active-contour segmentation tool
• Smaller average tree depth
- Manual corrections by expert on selected slices
► Oblique splits are superior and Figure 5: Comparison of oblique and
- Used for evaluation and validation only!
worth the additional computa- orthogonal split models w.r.t. average
tree depth (left) and out-of-bag (OOB)
tional effort during training
10
6
4
OSF
SFT
Feature class
Oblique Random Forests
• Efficient, scalable, and accurate predictor
• Oblique splits favorable over univariate orthogonal splits
- Better generalization in areas with few samples
- Smaller average tree depth (faster prediction)
• Superior to Hessian-based approaches
1
0
0
Steerable Filter Templates (SFT)
• Parameterization of OSF eigenfilters using Gaussian derivatives
• Preferable over OSF features for better problem modeling
- Explicit scale parameterization (multiscale nature of vessels)
- Efficient directional filtering (rotational invariant features)
• Efficient feature extraction: separability + steerability
1.5
0.5
2
Machine learning framework for 3-D vessel segmentation
Orthogonal
Oblique
2
OOB error [%]
Average tree depth
8
Conclusions
2.5
Orthogonal
Oblique
OSF
SFT
Feature class
Figure 7: Visualization of the segmented cerebrovascular network for a single axial slice (top)
and the entire 3-D test ROI (bottom) using different segmentation techniques. (a) Ground truth.
(b) Frangi [2]. (c) RF-OSF (d = 102). (d) RF-SFT (M = 4). The segmentation results are rendered in
error (right) for OSF and SFT features. 3-D (bottom) and outlined in red (top) along with the ground-truth contours in blue for three
Errorbars indicate the standard deviation subregions within the axial slice (A-C). Red contours in (a) mark the Otsu labels [8] used for RF
training. Black circles in the 3-D plots highlight prominent differences in the segmentation.
over the four datasets.
Acknowledgements
This work has been funded by the Swiss National Centre of Competence in Research on Computer Aided and Image Guided
Medical Interventions (NCCR Co-Me) funded by the Swiss National Science Foundation.
References
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pp. 130–137. Springer, Berlin/Heidelberg (1998)
[3] Freeman, W.T., Adelson, E.H.: The design and use of steerable filters. IEEE Trans Pattern Anal Mach Intell 13(9), 891–906 (Sep 1991)
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