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

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

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

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.

Can we extend this work to 3D orientation scores?

3D Orientation Scores

Processing via Orientation Scores Image Score Processed image Processed Score

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

Orientation Score Transformation Using Cake-Wavelets 2D 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

Construction Wavelets Separable wavelets in Fourier domain: Angular B-spline in SH-basis via pseudo-inv. of DISHT: arXiv nr. 1505.09670, 2015 F = Funk-transform Anti-symmetrize Rotate

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

Orientation Scores Image Score Processed image Processed Score

Overview Data-Adaptive Processing via Orientation Scores

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

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

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

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.