Neural network parametrized level sets for image segmentation

Abstract

The Chan-Vese functionals have proven to by a first-class method for segmentation and classification. Previously they have been implemented with level-set methods based on a pixel-wise representation of the level-sets. Later parametrized level-set approximations, such as splines, have been studied. In this paper we consider neural networks as parametrized approximations of level-set functions. We show in particular, that parametrized two-layer networks are most efficient to approximate polyhedral segments and classes. We also prove the efficiency for segmentation and classification.

Figures (11)

• Figure 1: Commutative diagram illustrating the relationships among the classical, parametrized, and smooth Chan–Vese models. The classical (unparametrized) model can be approximated by the parametrized non-smooth model, which can then be approximated by the smooth variant.
• Figure 2: One- and two-layer neural networks.
• Figure 3: 1. Each line $H_j^0 = \{{\tt a}_j = 0\}$, for $j=1,2,3$, divides the image domain into 2 half-planes $H_j^1$ and $H_j^{-1}$. The half-planes containing the triangle $T$ (shown in color) attain the value $1$ and the complementary half-planes (left uncolored) attain $0$. These 3 binary regions correspond to the activations of 3 neurons in the first layer of a neural network, represented by $\sigma({\tt a}_j)$, $j = 1, 2, 3$, where ${\tt a}_j > 0$ inside the region and ${\tt a}_j({\vec{x}}) < 0$ outside the region as the standard level-set method formulation. 2. Seven subregions $\Omega_{\iota}$ for $\iota \in (-1,1)^3$ and their triplets of activation values $(\sigma({\tt a}_1), \sigma({\tt a}_2), \sigma({\tt a}_3))$. 3. The average of these activations triplets are taken over each subregion $\Omega_\iota$, resulting in values $1/3,2/3,1$. This corresponds to the linear combination ${\tt s}_c = \sum_{j=1}^3 \frac{1}{3}\sigma({\tt a}_j)$. 4. Applying the activation $\sigma(-\frac{8}{9} + {\tt s}_c)$ to each subregion, only the region with ${\tt s}_c=1$ outputs the value $1$. This identifies the segmented regions $\Omega_{(1,1,1)} = T =: \Xi_1$. Conversely, applying the activation $-\sigma(-\frac{8}{9} + {\tt s}_c)+1$ to each subregion yields the complementary regions $\Xi_{-1} = \Omega\backslash\Xi_1$.
• Figure 4: Polygonal approximation of arbitrary bounded regions. Increasing the number of polygonal edges, which corresponds to increasing the number of neurons $n_1$ in the network, improves the approximation accuracy.
• Figure 5: Image segmentation characterized by level-set functions and their signs. Here, $1$ denotes the positive sign of a level-set function, corresponding to the inside of a region, and $-1$ denotes the negative sign, corresponding to the outside. Left: Segmentation with one level-set function $\ell$$(m=1)$, $\Omega$ is segmented in two segments $\Xi_{1}$ and $\Xi_{-1}$. Right: Segmentation with two level-set functions $\ell_1, \ell_2$$(m=2)$, $\Omega$ is segmented in four segments $\Xi_{\pm 1,\pm 1}$.

Theorems & Definitions (20)

• definition 2.1: Heaviside and sigmoid networks
• definition 2.2: Customized two-layer network
• example 2.3: Heaviside network of a triangle
• lemma 2.4
• proof
• remark 3.1
• definition 3.2: Multiphase level-set function
• remark 3.3
• example 3.4
• definition 3.5