In this paper, we propose a novel low dimensional manifold model (LDMM) and
apply it to some image processing problems. LDMM is based on the fact that the patch manifolds
of many natural images have low dimensional structure. Based on this fact, the dimension of the
patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve
a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The
point integral method enforces the sample point constraints correctly and gives better results than the
standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution
problems show that LDMM is a powerful method in image processing.
Inspired by the graph Laplacian and the point integral method, we introduce a novel weighted graph Laplacian method to compute a smooth interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image
inpainting show that the weighted graph Laplacian is a reliable and efficient interpolation method. In addition, it is easy to implement and faster than graph Laplacian.
In this paper we integrate semi-local patches and the weighted graph Laplacian into the framework of the low dimensional manifold model.
This approach is much faster than the original LDMM algorithm. The number of iterations is typically reduced from 100 to 10 and the equations in each step are much easier to solve. This new approach is tested in image inpainting and denoising and the results are better than the original LDMM and competitive with state-of-the-art methods.
The transformed $l_1$ penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm, where $p$ is in $(0,1)$. In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the $l_p$ thresholding functions ($p$ is in $[0,1]$) are in closed form only for $p=0;1;1=2;2=3$, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds (TL1IT-s1 for short), consistently performs the best in sparse signal recovery with and without measurement noise.
We present the application of a low dimensional manifold model (LDMM) on hyperspectral
image (HSI) reconstruction. An important property of hyperspectral images is that the
patch manifold, which is sampled by the three-dimensional blocks in the data cube, is generally of
a low dimensional nature. This is a generalization of low-rank models in that hyperspectral images
with nonlinear mixing terms can also fit in this framework. The point integral method (PIM) is used
to solve a Laplace-Beltrami equation over a point cloud sampling the patch manifold in LDMM.
Both numerical simulations and theoretical analysis show that the sample points constraint is correctly
enforced by PIM. The framework is demonstrated by experiments on the reconstruction of
both linear and nonlinear mixed hyperspectral images with a significant number of missing voxels
and several entirely missing spectral bands.