We introduce a new multi-dimensional nonlinear embedding-Piecewise Flat Embedding (PFE)-for image segmentation. Based on the theory of sparse signal recovery, piecewise flat embedding with diverse channels attempts to recover a piecewise constant image representation with sparse region boundaries and sparse cluster value scattering. The resultant piecewise flat embedding exhibits interesting properties such as suppressing slowly varying signals, and offers an image representation with higher region identifiability which is desirable for image segmentation or high-level semantic analysis tasks. We formulate our embedding as a variant of the Laplacian Eigen-map embedding with an L _1,p (0 <; p ≤ 1) regularization term to promote sparse solutions. First, we devise a two-stage numerical algorithm based on Bregman iterations to compute L _1,1 -regularized piecewise flat embeddings. We further generalize this algorithm through iterative reweighting to solve the general L _1,p -regularized problem. To demonstrate its efficacy, we integrate PFE into two existing image segmentation frameworks, segmentation based on clustering and hierarchical segmentation based on contour detection. Experiments on four major benchmark datasets, BSDS500, MSRC, Stanford Background Dataset, and PASCAL Context, show that segmentation algorithms incorporating our embedding achieve significantly improved results.

中文翻译：

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分段平面嵌入的图像分割

我们引入了一种新的多维非线性嵌入-Piecewise平面嵌入（PFE）-用于图像分割。基于稀疏信号恢复的理论，具有不同通道的分段平坦嵌入试图恢复具有稀疏区域边界和稀疏簇值散射的分段恒定图像表示。最终的分段平坦嵌入展现出令人感兴趣的特性，例如抑制缓慢变化的信号，并提供具有较高区域可识别性的图像表示，这对于图像分割或高级语义分析任务是理想的。我们将嵌入公式表示为Laplacian特征图嵌入的L _{1，p的变体} （0 <; p≤1）正则化项以促进稀疏解。首先，我们设计了一个基于Bregman迭代的两阶段数值算法，以计算L _1,1 正则化的分段平面嵌入。我们通过迭代重加权来进一步推广该算法，以解决一般的L _1，p 正则化问题。为了证明其有效性，我们将PFE集成到两个现有的图像分割框架中，即基于聚类的分割和基于轮廓检测的分层分割。对四个主要基准数据集BSDS500，MSRC，斯坦福背景数据集和PASCAL Context进行的实验表明，结合了我们的嵌入效果的细分算法可显着改善结果。