# DAVIDSTUTZ

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19thMARCH2015

C. Conrad, M. Mertz, R. Mester. Contour-relaxed superpixels. Energy Minimization Methods in Computer Vision and Pattern Recognition, 2013.

This paper introduces Contour Relaxed Superpixels, a statistical approach to superpixel segmentation. In particular, the value $I_c(x_n)$ of pixel $x_n$ in channel $c$ is assumed to be the outcome of stochastic process described by the parameters $\theta_{s(x_n), c}$ of the corresponding superpixel $S_{s(x_n)}$ where $s(x_n)$ denotes the superpixel $x_n$ belongs to. Using $\theta$ to denote the set of all such parameters, the superpixel segmentation $S$ maximizing $p(S, \theta | I)$ is searched for. Using bayes theorem, and omitting the normalization factor, the energy to be maximized is given by

$p(S, \theta | I) = \frac{p(I | S, \theta) p(S, \theta)}{p(I)} \propto p(I |S, \theta) p(S, \theta) =: E(S)$.

The parameters $\theta$ are considered deterministic parameters such that $p(S, \theta)$ can be simplified to $p(S, \theta) = \kappa p(S)$. An EM-style (e.g. see [1, p. 423ff]) algorithm is applied: The parameters $\theta$ are optimized using maximum likelihood considering the superpixel segmentation to be constant followed by optimizing for $S$ while the parameters $\theta$ are held constant. Considering each pixel connected to $8$ neighbors, $p(S)$ is modeled using a Gibbs Random Field and can be factorzed into

$p(S) = \kappa' \exp(-N_e C_e - N_v C_v)$

where only the second factor depends on the label of the pixel $x_n$. Here, $N_e$ is the number of direct neighbors of $x_n$ having a different label and $N_v$ is the number of diagonal neighbors with a different label ‐ $C_e$ and $C_v$ are the associated costs. Furtermore, the probability $p(I | S, \theta)$ can be factorized as

$p(I | S, \theta) = \prod _{S_i \in S}\prod_{x_n \in S_i} \prod_{c=1}^3 p(I_c(x_n) | \theta_{i,c})$

where all the stoachstic processes are considered independent.

The optimization is carried out as follows: Iteratively, each boundary pixel $x_n$ is considered to change its label. When $\theta$ is assumed constant, $x_n$ is assigned the label maximizing

$E(S) = \kappa \kappa' \exp(-N_e C_e - N_v C_v) \prod_{S_i} \prod_{x_m \in S_i}\prod_{c=1}^3 p(I_c(x_m)|\theta_{i,c})$.(1)

This approach can be categorized as gradient ascent approach to superpixel segmentation and is summarized in algorithm 1. The first product in equation (1) runs over all superpixels $S_i$ to which pixel $x_n$ may be assigned.

function crs(
$I$, // Color image.
$K$ // Number of superpixels.
)
// The step size R can be derived from the image size W ×H and K:
initialize $S$ as regular grid with step size $R$
initialize $\theta$ using sufficient statistics (e.g. Gaussian)
for $t = 1$ to $T$
// Originally, the image is traversed multiple times using different
// directions to avoid a directional bias:
for $n = 1$ to $N$
if $x_n$ is a boundary pixel
// This can be evaluated by taking θ as constant; Conrad et al.
// suggest to minimize the negative logarithm of (1) instead:
assign $x_n$ to the label maximizing equation (1)
return $S$

Algorithm 1: Countour relaxed superpixels as iterative energy maximization.

A C++ implementation of Contour Relaxed Superpixels is available at the project's webpage. Superpixel segmentations generated using the described approach are shown in figure 1.

• [1] C. Bishop. Pattern Recognition and Machine Learning. Springer Verlag, New York, 2006.
• [2] P. Arbeláez, M. Maire, C. Fowlkes, J. Malik. Contour detection and hierarchical image segmentation. Transactions on Pattern Analysis and Machine Intelligence, volume 33, number 5, pages 898–916, 2011.

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