Normalized Cuts and Image Segmentation

Abstract
Model image segmentation problem by normalized cuts, then solve the relaxed normalized cuts problem using spectral clustering.

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Notations

$G$: $G=(V,E)$ be an undirected graph with vertex set $V=\{v_1,\dots,v_n\}$.
$W$: The adjacency matrix of $G$, we assume each edge between two vertices $v_i$ and $v_j$ carries a non-negative weight $w_{ij}\geq 0$.
$d_i$: The degree of a vertex $v_i\in V$, which is define as $d_i=\sum\limits_{j=1}^n{w_{ij}}$.
$D$: $=diag(d_i), i=1,\dots n$.
$L$: Graph Laplacian of $G$, which is defined as $L=D-W$
$L_{sym}$: $=D^{-\frac{1}{2}}LD^{-\frac{1}{2}}$
$\text{W}(A_1, A_2)$: $=\sum\limits_{v_i \in V_1, v_j \in V_2}{w_{ij}}$, where $A_1, A_2$ are disjoint subsets of $V$.
$\text{vol}(A_1)$: $=\sum\limits_{v_i\in A_1}{d_i}$, where $A_1 \subset V$.
$Ncut(A_1,\dots, A_s)$: The normalized cut of