  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 Eigenmap embedding with
an $L_{1,p} (0<p\leq1)$ 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.
