Total Bregman divergence and its applications to DTI analysis

Abstract

Divergence measures provide a means to measure the pairwise dissimilarity between "objects," e.g., vectors and probability density functions (pdfs). Kullback-Leibler (KL) divergence and the square loss (SL) function are two examples of commonly used dissimilarity measures which along with others belong to the family of Bregman divergences (BD). In this paper, we present a novel divergence dubbed the Total Bregman divergence (TBD), which is intrinsically robust to outliers, a very desirable property in many applications. Further, we derive the TBD center, called the t-center (using the l(1)-norm), for a population of positive definite matrices in closed form and show that it is invariant to transformation from the special linear group. This t-center, which is also robust to outliers, is then used in tensor interpolation as well as in an active contour based piecewise constant segmentation of a diffusion tensor magnetic resonance image (DT-MRI). Additionally, we derive the piecewise smooth active contour model for segmentation of DT-MRI using the TBD and present several comparative results on real data.

Fig. 1

In each figure, d_f(x, y) (dotted line) is BD, δ_f(x, y) (bold line) is TBD. Left figure shows d_f(x, y) and δ_f(x, y) before rotating the coordinate system. Right figure shows d_f(x, y) and δ_f(x, y) after rotating the coordinate system. d_f(x, y) changes with rotation while δ_f(x, y) is invariant with rotation.

Fig. 2

The isosurfaces of d_F (P, I) = r, d_R (P, I) = r, KL_s (P, I) = r and tKL (P, I) = r shown from left to right. The three axes are eigenvalues of P.

Fig. 3

From left to right are initialization, intermediate step, and final segmentation.

Fig. 4

Dice coefficient comparison for tKL, KL_s, d_R, d_M, and LE segmentation of synthetic tensor field with increasing level (x-axis) of noise.

Fig. 5

Dice coefficient comparison for tKL, KL_s, d_R, d_M, and LE segmentation of synthetic tensor field with increasing percentage (x-axis) of outliers.

Fig. 6

(a)–(c) Initialization. (d) Segmentation results, from left to right, using tKL, KL_s, d_R, d_M, and LE.

Fig. 7

tKL segmentation of a 3-D rat corpus callosum. (a)–(d) A 2-D slice of the corresponding evolving surface, from left to right are initialization, intermediate steps and final segmentation. (e) A 3-D view of the segmentation result.