Sharpening Transform diffusion tensor into fiber tensor Improves

Sharpening Transform diffusion tensor into fiber tensor Improves tracking based on the full tensor

Problem with full tensor tracking algorithms • • Diffusive tracking Leaking of the tracts in unexpected regions

What is done in the literature? • Take power of the diffusion tensor D [ n = 7 Koch et al Neuro. Image 2002, n = 4 Tournier et al Neuro. Image 2003, n = 2 Lazar et al HBM 2003] • • Heuristic Create degenerate tensors

A new sharpening transform • Theoretically sound: • Deconvolution by the response function of a single fiber • Simple linear transformation when using spherical harmonics • Transforms diffusion tensor into fiber tensor • Natural pre-processing tool for fiber tracking

Sketch of the approach Fiber tensor Diffusion tensor D Deconvolution on the sphere by fiber response function Spherical function, x. Dx. T Sharp spherical function

Step 1: Diffusion tensor to spherical function • Given diagonal diffusion tensor D • The quadratic form of D is a function on the sphere (x y z) D (x y z)T = ax 2 + by 2 + cz 2

Sketch of the approach Diffusion tensor D Spherical function, x. Dx. T

Step 2: Spherical harmonic estimation of spherical function • Spherical harmonics (SH) form an orthonormal basis for all complex functions on the sphere • Define basis of spherical harmonics, Yj, that is real and symmetric • • Symmetry: choose only even order SH Real: take real part of SH

Spherical harmonic description of the spherical function • Given spherical function S( ) • • In our case, S is the diffusion tensor on the sphere Find the coefficients, cj, such that where R is the number of spherical harmonics in the basis of order L [Descoteaux et al, MRM 2006]

Sketch of the approach Diffusion tensor D Spherical harmonics description of the signal S c 1, c 2, …. , c. R Spherical function, S = x. Dx. T

Step 3: Sharpening • • Deconvolution of the spherical function, S( ), with response function of a single fiber, R( ) Inspired by Tournier et al Neuro. Image 2004 • Spherical deconvolution to obtain the fiber orientation distribution function F( )

Sharpening as a deconvolution => Need to define good Rsharp

Defining fiber response function Rsharp • • Assume a Gaussian can describe the diffusion of H 20 for a single fiber Can change variance of the Gaussian to obtain the desired sharpening factor k k=1 k=3

Deconvolution as a simple matrix multiplication • If S is expressed in SH basis, one can use the Funk-Hecke theorem to solve the integral analytically Deconvolution done with a matrix multiplication: linear transformation of the SH coefficients Ssharp = S * 1/Rsharp

Sketch of the approach Diffusion tensor D Deconvolution on the sphere by fiber response function Ssharp = S * 1/Rsharp Spherical function, S = x. Dx. T Sharp spherical function Ssharp

Step 4: Spherical harmonics coefficients to tensor elements • Even order spherical harmonics up to order l and rank-l tensors restricted to the sphere are bases for the same function space. Proof in [Descoteaux et al, MRM 2006] • Simple linear operation to go from sharp spherical function in SH basis to fiber tensor

Resulting sharpening: diffusion tensor to fiber tensor One-to-one linear transformation from tensor basis to spherical harmonic basis [Descoteaux et al, MRM 2006] DT Diffusion tensor SH sharp SH fiber DT

Methods 1. Data Acquisition: 60 directions, b = 1000 s/mm 2 2. Compute diffusion tensors with least-squares 3. Transform diffusion tensor into fiber tensor 4. Geodesic Connectivity Mapping

GCM • • • Bla bla Connectivity …

Results Less degenerate tensors Improves tracking

Degenerate tensors 1. 2. Deconvolution sharpening does not introduce degenerate tensors Power method introduces very small and zero eigen values rapidly

Splenium of the Corpus Callosum (CC) Diffusion tensor Fiber tensor Average Connectivity Index Minimum

Splenium of the CC: different initialization start Diffusion tensor start Fiber tensor

Splenium of the CC Diffusion tensor Fiber tensor

Anterior Thalamic Radiations (atr) “gold” Standard atr [Mori et al atlas 05]

Anterior Thalamic Radiations (atr) start Diffusion tensor start Fiber tensor

Anterior Thalamic Radiations (atr) Diffusion tensor Fiber tensor

Anterior Thalamic Radiations (atr) Axial slice Sagittal Slice [Koch et al, Neuro. Image 2002, Anwander et al, Cerebral Cortex 2007]

Cortico spinal tract (cst) “gold” Standard atr

Cortico spinal tract (cst) start Diffusion tensor Fiber tensor

Cortico spinal tract (cst) Diffusion tensor Fiber tensor

Cortico spinal tract (cst) Axial slice Sagittal Slice [Koch et al, Neuro. Image 2002, Anwander et al, Cerebral Cortex 2007]

Stria terminalis (st) “gold” standard st [Mori et al atlas 05]

Stria terminalis (st) start Diffusion tensor leaking

Stria terminalis (st) start Diffusion tensor No leaking

Stria terminalis (st) leaking Missing the most curving fibers

Stria terminalis (st) • • Picking up the most curving part Missing the fiber part under the corpus callosum [Koch et al, Neuro. Image 2002, Anwander et al, Cerebral Cortex 2007]