Subspace learning for Mumford&#x2013;Shah-model-based texture segmentation through texture patches

Yan Nei Law; Hwee Kuan Lee; Andy M. Yip

doi:10.1364/AO.50.003947

Applied Optics
Vol. 50,
Issue 21,
pp. 3947-3957
(2011)
•https://doi.org/10.1364/AO.50.003947

Subspace learning for Mumford–Shah-model-based texture segmentation through texture patches

Yan Nei Law, Hwee Kuan Lee, and Andy M. Yip

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Yan Nei Law, Hwee Kuan Lee, and Andy M. Yip, "Subspace learning for Mumford–Shah-model-based texture segmentation through texture patches," Appl. Opt. 50, 3947-3957 (2011)

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Abstract

In this paper, we develop a robust and effective algorithm for texture segmentation and feature selection. The approach is to incorporate a patch-based subspace learning technique into the subspace Mumford–Shah (SMS) model to make the minimization of the SMS model robust and accurate. The proposed method is fully unsupervised in that it removes the need to specify training data, which is required by existing methods for the same model. We further propose a novel (to our knowledge) pairwise dissimilarity measure for pixels. Its novelty lies in the use of the relevance scores of the features of each pixel to improve its discriminating power. Some superior results are obtained compared to existing unsupervised algorithms, which do not use a subspace approach. This confirms the usefulness of the subspace approach and the proposed unsupervised algorithm.

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Rank	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22–32
ER	455	30	12	9	1	1	0	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0
WMV	377	65	23	10	5	3	4	4	5	3	3	5	0	2	1	1	0	0	0	0	1	0

Image	Auto-SMS	MS	${MS}_{NLST}$	k-means
RGB	0.4%	22.9%	40.4%	38.1%
Tissue	9.8%	13.8%	15.8%	14.4%
Stem cell	6.5%	6.8%	35.9%	16.6%
Cuttlefish	5.3%	21.7%	14.1%	20.3%

1:	Input: image u
2:	Extract features ${f_{j}}_{j = 1}^{m}$ from u
3:	Select k random pixels $S = {x_{1}, \dots, x_{k}}$
4:	For each $x \in S$ , determine the size r of the local patch $R_{x}$ around x
5:	Cluster the patches ${R_{x}}_{x \in S}$ into n groups ${S_{i}}_{i = 1}^{n}$
6:	Optimize the objective $F^{Auto}$ using Algorithm 2
7:	Output: segmentation ${Ω_{i}}$

1:	Input: features ${f_{j}}$ , a clustering of patches ${S_{i}}$
2:	Initialize $β^{0} = 1$
3:	Initialize ${Ω_{i}^{0}}$ using, e.g., k-means
4:	Initialize $k = 0$
5:	repeat
6:	${c_{i j}^{k + 1}} = {argmin}_{{c_{i j}}} F^{Auto} ({Ω_{i}^{k}}, {c_{i j}}, {λ_{i j}^{k}})$
7:	${λ_{i j}^{k + 1}} = {argmin}_{{λ_{i j}}} F^{Auto} ({Ω_{i}^{k}}, {c_{i j}^{k + 1}}, {λ_{i j}})$
8:	${Ω_{i}^{k + 1}} = {argmin}_{{Ω_{i}}} F^{Auto} ({Ω_{i}}, {c_{i j}^{k + 1}}, {λ_{i j}^{k + 1}})$ (by Algorithm 3 in Appendix A)
9:	if k is a multiple of K then $β^{k + 1} = 0.9 β^{k}$ else $β^{k + 1} = β^{k}$ endif
10:	$k \leftarrow k + 1$
11:	until ${Ω_{i}^{k}}$ stabilizes
12:	Output: segmentation ${Ω_{i}^{k}}$

1:	Input: features ${f_{j}}$ , weights ${λ_{i j}}$ , fitting constants ${c_{i j}}$
2:	Initialize $u^{0}$ by using k-means or the $u^{k}$ from the last run of Algorithm 3 or a random assignment
3:	Initialize $p^{0} (x) = (0, 0)$ for each pixel x
4:	Initialize $k = 0$
5:	Set $r (x) = \sum_{j = 1}^{m} {λ_{1 j} [c_{1 j} - f_{j} (x)]^{2} - λ_{2 j} [c_{2 j} - f_{j} (x)]^{2}}$ for each $x \in Ω$
6:	repeat
7:	for each pixel $x \in Ω$ do
8:	$χ^{k + 1} (x) = {\begin{cases} 1 & if u^{k} (x) > 0.5, \\ 0 & otherwise \end{cases}$ $v^{k + 1} (x) = \min {\max {u^{k} (x) - θ r (x), 0}, 1}$ $p^{k + 1} (x) = \frac{p^{k} (x) + δ [\nabla (div p^{k} - \frac{v^{k + 1}}{μ θ})] (x)}{1 + δ \| [\nabla (div p^{k} - \frac{v^{k + 1}}{μ θ})] (x) \|}$ $u^{k + 1} (x) = v^{k + 1} (x) - μ θ (div p^{k + 1}) (x) .$
9:	end for
10:	$k \leftarrow k + 1$
11:	until $u^{k}$ converges
12:	Output: segmentation $χ^{k}$

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Applied Optics