1. Invertible Orientation Scores of 3D Images
Michiel Janssen,
Remco Duits & Marcel Breeuwer
Dep. of Mathematics and Computer Science & Dep. of BME TU/e Eindhoven
Project: ERC Lie Analysis

2. Motivation Elongated structures appear in medical images
Methods for detection and enhancement often fail at crossings/bifurcations etc.
Retina
Muscle Cells
Vessels in brain

3. Orientation Scores - Background
To cope with crossings/bifurcations orientation scores (OS) were introduced
In the OS crossing structures are disentangled
Image
Filter
Orientation Score
Orientation y x
Image OS
Image OS

4. Orientation Scores - Applications
2D Retinal Vessel Tracking in Orientation Scores
Crossing-Preserving Coherence Enhancing Diffusion via Invertible Orientation Scores
Image OS
Noise Reduction
Franken & Duits IJCV 2008
Duits & Franken QAM AMS part I &II 2010
With multiple scale OS: Sharma & Duits ACHA 2014
- Bekkers & Duits et al., JMIV 2014.
- Bekkers, Duits, Mashtakov, Sanguinetti
SSVM 2015, arxiv to SIAM SIIMS.

5. Can we extend this work to 3D orientation scores?

6. 3D Orientation Scores

7. Processing via Orientation Scores
Image
Score
Processed
image
Processed
Score

8. Reconstruction
1. Exact Reconstruction by adjoint of unitary map on reproducing kernel space
2. Approximate reconstruction by adjoint:
3. Approximate fast reconstruction:
All frequencies covered
Well posedness is Quantified by M psi
Lower bound and upper bound of m psi determine the condition number

9. Orientation Score Transformation Using Cake-Wavelets2D
Spectrum covered
Stable reconstruction
& all scales merged.
Inverse
DFT
Anti-symmetrize 3D
All frequencies covered
Well posedness is Quantified by M psi
Lower bound and upper bound of m psi determine the condition number
Funk-Transform
Inverse DFT

10. Construction Wavelets
Separable wavelets in Fourier domain:
Angular B-spline in SH-basis via pseudo-inv. of DISHT:
arXiv nr , 2015
F = Funk-transform
Anti-symmetrize
Rotate

11. Fully Analytic Wavelet Design via 3D Generalized Zernike Basis
For controlling wavelet shape and stability of reconstruction we need analytical description in both domains.
IJCV 2007: Spectral decomposition of harmonic oscillator:
Recent with Guido Janssen: Expansion in 3D Gen. Zernike basis:
& multiple scale extensions…
oscillations
- In box pol of degree n. Weight (1-\rho^2)^{-\alpha}. Nice formulas for Fourier transform, Radon transform, and recursion formulas, scaling relations.
- P_k^{\alpha,\beta} Jacobi pol. Weight (1-x)^\alpha(1+x)^{\beta}
- Note m=-l..ln=l+2p

12. Orientation Scores
Image
Score
Processed
image
Processed
Score

13. Overview Data-Adaptive Processing via Orientation Scores

14. Exponential Curve Fits to 3D OS in SE(3)
Eigenvector Structure Tensor on SE(3)
Eigenvector Hessian on SE(3)
Tangent vector of local, 1st order exponential curve fit
Tangent vector of local, 2nd order exponential curve fit
Spatial
Projections
Induces unique locally adaptive gauge frame:
For Theorems see Duits and Janssen et al. Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging, to JMIV, see Arxiv, 2015

15. Data-Adaptive Diffusions via Invertible Orientation Scores of 3D images
Processed
Score
Processed
image
Not adaptive using
Image
Adaptive using

16. Application in 3D Vessel Analysis
Vesselness
Vessel Segmentation
Vesselness via OS
(preserves bifurcations)
Vesselness
Next Challenge :

17. Conclusion Main Results Future work Related Result
3D Cake-Wavelets for invertible OS:
Crossing Preserving Diffusion (CEDOS)
generalized to 3D.
Related Result
Locally adaptive frames
(via exp. curve fits) Duits & Janssen et al., see Arxiv, 2015.
Future work
Sub-Riemannian wavefront propagation in SE(3).
Application in Detection 3D vessels.
Application in fiber tracking/enhancement in DW-MRI.