Texture Edge detection by Patch consensus (TEP)

TL;DR

This work proposes a new simple way to identify the texture edge location, using the consensus of segmented local patch information, and derives the necessary condition for textures to be distinguished, and analyze the patch width with respect to the scale of textures.

Abstract

We propose Texture Edge detection using Patch consensus (TEP) which is a training-free method to detect the boundary of texture. We propose a new simple way to identify the texture edge location, using the consensus of segmented local patch information. While on the boundary, even using local patch information, the distinction between textures are typically not clear, but using neighbor consensus give a clear idea of the boundary. We utilize local patch, and its response against neighboring regions, to emphasize the similarities and the differences across different textures. The step of segmentation of response further emphasizes the edge location, and the neighborhood voting gives consensus and stabilize the edge detection. We analyze texture as a stationary process to give insight into the patch width parameter verses the quality of edge detection. We derive the necessary condition for textures to be distinguished, and analyze the patch width with respect to the scale of textures. Various experiments are presented to validate the proposed model.

Figures (18)

• Figure 1: Challenges of texture edge detection. (a) An given image with textures. (b) Canny edge detection. (c) Chan-Vese segmentation.
• Figure 2: Texture Edge detection by Patch consensus (TEP). For each patch $\vec{\mathcal{P}}(\mathbf{x})$ in the given image $U$, the patch response $\mathcal{R}(\mathbf{y};\mathbf{x})$ is computed. The patch response is segmented, and the boundary of the phases gives the local edge $\mathcal{W}(\mathbf{y},\mathbf{x})$. The edge function $V(\mathbf{x})$ is computed by the consensus of the local edge function $\mathcal{W}(\mathbf{y},\mathbf{x})$.
• Figure 3: (a) Synthetic texture image with two textures $\mathcal{P}$ (left) and $\mathcal{Q}$ (right). Two patches $\vec{\mathcal{P}}(\mathbf{x})$ and $\vec{\mathcal{Q}}(\mathbf{y})$ are marked with blue and yellow. (b) and (c) show two patch responses $\mathcal{R}(\cdot; \mathbf{x})$ and $\mathcal{R}(\cdot; \mathbf{y})$ respectively. Note that the texture edge is clearly emphasized with a suitable patch width parameter $r$.
• Figure 4: (a) The intensity histograms of two textures in Figure \ref{['fig:analysis_experiment']} (a). (b) Estimated distributions of the patch response $\mathbb{E}_{\mathbf{y}|\mathbf{x}}\left(\mathcal{R}(\mathbf{y}; \mathbf{x})\right)$ as $r$ increases, assuming the patch centered at $\mathbf{x}$ is sampled from $\mathcal{P}$, and $\mathbf{y}$ from $\mathcal{P}$ or $\mathcal{Q}$ as indicated in the legend. (c) Same as (b) assuming the patch centered at $\mathbf{x}$ is sampled from $\mathcal{Q}$.
• Figure 5: Change of distance of distribution function of $\mathbb{E}_{\mathbf{y}|\mathbf{x}}(\mathcal{R}(\mathbf{y};\mathbf{x}))$ with respect to the patch width parameter $r$. Blue line assumes the observer $\mathbf{x}$ being equipped with $\mathcal{P}$, while the red line assumes $\mathbf{x}$ being equipped with $\mathcal{Q}$. The horizontal dashed line indicates the required minimal distance for two textures to be distinguished by the segmentation model.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1: Expectation of quadratic form seber2003linear
• Theorem 1
• Theorem 2
• proof
• proof