Use of Particle Swarm Optimization for ODF maxima extraction

Fiber tracking is winning more and more interest in the neuroscience research field and clinical practice, for its ability in revealing the structural connectivity; the quality of the fiber tracking depends in great extent, on fiber directions extraction The PSO algorithm could give good approximation of these directions. I. MATERIALS AND METHODS The diffusion tensor MRI (DTI) was introduced by Basser et al. 1994 [1], it captures and quantifies the free or constrained water molecules movement; symmetric positive definite second order diffusion tensors were used to model the profile of the diffusion for each voxel, with the assumption of a single bundle fiber per voxel. Thus, tractography algorithms based on DTI may produce unreliable results. To overcome this limitations, new methods and techniques of High Angular Resolution Diffusion Imaging (HARDI) [2] have been proposed: the Q-ball imaging (QBI), [3] with a spherical sampling of the diffusion space, Diffusion Spectrum Imaging (DSI) with a sampling of the entire Cartesian grid 3D space diffusion, spherical deconvolution techniques. These techniques allow the reconstruction of the multifiber by calculating probability density function (PDF) which is estimated by the Orientation Density Function (ODF) whose maxima are aligned with the actual directions of the fibers. The importance of tractography in clinical studies makes the ODF maxima extraction a crucial post-processing step, since the ODF provides the angular information by having its maxima aligned on the underlying fiber directions. In the present work, we propose a new ODF maxima search approach using the algorithm Particle Swarm Optimization (PSO), to efficiently and accurately extract all the fibers directions. PSO introduced by Kennedy and Eberhart [4] in 1995, is an optimization technique of adaptive research based on population. Through a trial and error process [5], PSO assigns a randomized velocity to each potential solution, called particle, and then fly through the problem space. A. The PSO Algorithm Each particle represents a potential solution in the search space. The new position of a particle is determined by its own value and that of its neighbors. In the algorithm xi ⃗⃗⃗ (t) is the position of the particle Pi, at instant t, which changes by adding a velocity vi ⃗⃗⃗ (t) to its current position. vi(t) is the particle i velocity at the instant t and xi(t) is the position of the particle i at instant t, parameters: inertia (w), learning coefficients c1, and c2 are constants set by the user, ( 0 ≤ w ≤ 1.2 , 0 ≤ c1 ≤ 2 and 0 ≤ c2 ≤ 2) [6] coefficients r1 and r2 are random numbers drawn at each iteration, g(t) is the best solution found so far and t xpi(t) is the best solution found by the particle Pi; in the following algorithm N is the number of particles; pbesti Best fitness obtained for the particle PI; xpbesti ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ Position of the particle Pi for her best fitness; xgbesti ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ Position of the particle with the best fitness of all. PSO Algorithm Begin Repeat For i from 1 to N do If (F(xi ⃗⃗⃗ ) > pbestihen pbesti ← F(xi ⃗⃗⃗ ) xpbesti ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ← xi ⃗⃗⃗ End If If (F(xi ⃗⃗⃗ ) > gbesti) Then gbesti ← F(xi ⃗⃗⃗ ) xgbesti ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ← xi ⃗⃗⃗ End If End For For i de 1 à N faire vi ⃗⃗⃗⃗ ← vi ⃗⃗⃗ + r1c1(xpbesti ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ − xi ⃗⃗⃗ ) + r2c2(xgbesti ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ − xi ⃗⃗⃗ ) xi ⃗⃗⃗⃗ ← xi ⃗⃗⃗ + vi ⃗⃗⃗ End For Until (the process converges). End To optimize the algorithm and find the maxima efficiently, we evaluated the behavior of PSO according to different values of the constant parameters: inertia and learning coefficients (c1 and c2). We have chosen to minimize four functions of dimension 2: Two functions: Rosenbrock and Zakharov functions with a single minimum and Shafer and Himmelblau functions with multi-minimum. B. PSO parameterization a. Inertia To evaluate the effect of inertia we set learning factors at c1 = c2 = 2, and do five tests for each value of inertia (w). The results of these tests are represented on fig. 1, for different values of w between 0 and 1. For functions with only one minimum, the search for the optimum is achieved efficiently by using fixed values of inertia below 0.8. To deal with multi-minima problems, the best result is obtained when Use of Particle Swarm Optimization for ODF maxima extraction Mouloud KACHOUANE-Non Member, Thinhinane MEGHERBI-Non Member, Fatima OULEBSIRBOUMGHAR-EMBS Member, and Rachid DERICHE-EMBS Member the inertia is variable and decreases between 0.9 and 0.4 in each iteration. b. Learning coefficients We conducted other tests to evaluate the effect of learning factor, w = 0.8 for first two functions and decreasing from 0.9 to 0.4 for the others. Table I illustrates the success rates for four benchmark functions when the learning factors, c1 and c2, are both set from 1 to 3. For single minimum functions (table 1.a and 1.b), the algorithm is efficient when the sum of the two coefficients is in the vicinity of four. In case of multi-minimum, PSO loss effectiveness for high values of learning factors. C. Orientation Distribution Function (ODF) The ODF function is estimated and expressed in the spherical harmonics basis: S(θi , φi) = ∑ cj Yj(θi , φi) R j=1 (3) Yj are the spherical harmonics components, θ and φ are the spherical coordinates. Since the spherical function ODF is assumed to be real and antipodally symmetric, a modified version of this basis, real symmetric spherical harmonics, has been introduced in [4] and given by the following expressions: