Numerical Conditioning Problems and Solutions for Nonparametric i.i.d. Statistical Active Contours

Abstract

In this paper, we propose an active contour model based on nonparametric independent and identically distributed (i.i.d.) statistics of the image that can segment an image without any a priori information about the intensity distributions of the region of interest or the background. This is not, however, the first active contour model proposed to solve the segmentation problem under these same assumptions. In contrast to prior active contour models based on nonparametric i.i.d. statistics, we do not formulate our optimization criterion according to any distance measure between estimated probability densities inside and outside the active contour. Instead, treating the segmentation problem as a pixel-wise classification problem, we formulate an active contour to minimize the unbiased pixel-wise average misclassification probability (AMP). This not only simplifies the problem by avoiding the need to arbitrarily select among many sensible distance measures to measure the difference between the probability densities estimated inside and outside the active contour, but it also solves a numerical conditioning problem that arises with such prior active contour models. As a result, the AMP model exhibits faster convergence with higher accuracy and robustness when compared to active contour models previously formulated to solve the same nonparametric i.i.d. statistical segmentation problem via probability distances. To discuss this improved numerical behavior more precisely, we introduce the notion of "conditioning ratio" and demonstrate that the proposed AMP active contour is numerically better conditioned (i.e., exhibits a much smaller conditioning ratio) than prior probability distance-based active contours.

An Efficient MRF Embedded Level Set Method for Image Segmentation

“…All experiments are performed on MATLAB R2010b installed in a computer with a 3.30GHz Intel Core i3 CPU and 4GB of RAM. The reference methods are GAC [6], DAC [9], CV [14], LBF [15], AMP [16], TCF [17], GM [25], TS [26] and GC [27], briefed in the Introduction section. Note the comparison methods and our method are pure data-driven and do not take the advantages of the shape priors.…”

“…Existing level set methods for image segmentation can be grouped into two categories: edge-based models [6][7][8][9][10][11][12][13] and region-based models [14][15][16][17][18].…”

“…Li et al [15] solved the problem of segmenting images with intensity inhomogeneity by using a local binary fitting (LBF) energy. By minimizing the unbiased pixel-wise average misclassification probability (AMP), Yezzi et al [16] formulated an active contour to segment an image without any prior information about the intensity distribution of regions. By realizing curve evolution via simple operations between two linked lists, Shi et al [17] achieved a fast level set algorithm for real-time tracking.…”

Quantitative measurement of pseudoexfoliation in the anterior segment of the eye performed in visible light

“…These include the active contour method [18], morphological methods or even simple binarization methods. It is also possible to use and profile well-known automatic methods of image analysis and processing from other branches of medicine [19-21].…”

An active contour model based on local fitted images for image segmentation

“…The minimization of energy functional E ( ϕ ) is equal to search for the steady-state solution of the following gradient flow formulation: $$\frac{\partial \varphi }{\partial t}=-\frac{\partial E}{\partial \varphi }$$ where $\frac{\partial E}{\partial \varphi }$ is the Gateaux derivative [37,38,43] of E ( ϕ ) with respect to ϕ . In our proposed model, the derivations of regularization terms of ϕ in Eq.…”