We construct a highly regular and simple structured class of sparsely connected convolutional neural networks with rectifier activations that provide universal function approximation in a coarse-to-fine manner with increasing number of layers. The networks are localized in the sense that local changes in the function to be approximated only require local changes in the final layer of weights. At the core of the construction lies the fact that the characteristic function can be derived from a convolution of characteristic functions at the next coarser resolution via a rectifier passing. The latter refinement result holds for all higher order univariate B-splines.

The blind source separation (BSS) problem extracts unknown sources from observations of their unknown mixtures. A current trend in BSS is the semiblind approach, which incorporates prior information on sources or how the sources are mixed. The constrained independent component analysis (ICA) approach has been studied to impose constraints on the famous ICA framework. We introduced an alternative approach based on the null space component (NCA) framework and referred to the approach as the c-NCA approach. We also presented the c-NCA algorithm that uses signal-dependent semidefinite operators, which is a bilinear mapping, as signatures for operator design in the c-NCA approach. Theoretically, we showed that the source estimation of the c-NCA algorithm converges with a convergence rate dependent on the decay of the sequence, obtained by applying the estimated operators on corresponding sources. The c-NCA can be formulated as a deterministic constrained optimization method, and thus, it can take advantage of solvers developed in optimization society for solving the BSS problem. As examples, we demonstrated electroencephalogram interference rejection problems can be solved by the c-NCA with proximal splitting algorithms by incorporating a sparsity-enforcing separation model and considering the case when reference signals are available.

From the perspective of the Bayesian approach, the denoising problem is essentially a prior probability modeling and estimation task. In this paper, we propose an approach that exploits a hidden Bayesian network, constructed from wavelet coefficients, to model the prior probability of the original image. Then, we use the belief propagation (BP) algorithm, which estimates a coefficient based on all the coefficients of an image, as the maximum-a-posterior (MAP) estimator to derive the denoised wavelet coefficients. We show that if the network is a spanning tree, the standard BP algorithm can perform MAP estimation efficiently. Our experiment results demonstrate that, in terms of the peak-signal-to-noise-ratio and perceptual quality, the proposed approach outperforms state-of-the-art algorithms on several images, particularly in the textured regions, with various amounts of white Gaussian noise.

A number of techniques have been developed to deal with image denoising, which is regarded as the simplest inverse problem. In this paper, we propose an approach that constructs a Bayesian network from the wavelet coefficients of a single image such that different Bayesian networks can be obtained from different input images. Then, we utilize the maximum-a-posterior (MAP) estimator to derive the wavelet coefficients. Constructing a graphical model usually requires a large number of training images. However, we demonstrate that by using certain wavelet properties, namely, interscale data dependency, decorrelation between wavelet coefficients, and sparsity of the wavelet representation, a robust Bayesian network can be constructed from one image to resolve the denoising problem. Our experiment results show that, in terms of the peak-signal-to-noise-ratio (PSNR) performance, the proposed approach outperforms state-of-art algorithms on several images with various amounts of white Gaussian noise.

Inspired by Paragious and Deriche's work, which unifies boundary-based and region-based image partition approaches, we integrate the snake model and the Fisher criterion to capture, respectively, the boundary information and region information of microarray images. We then use the proposed algorithm to segment the spots in the microarray images, and compare our results with those obtained by commercial software. Our algorithm is automatic because the parameters are adaptively estimated from the data without human intervention.

Inspired by Paragious and Deriche's work, which unifies boundary-based and region-based image partition approaches, we integrate the active contour(snake) model and the Fisher criterion to capture, respectively, the boundary and region information of microarray images. We then use the proposed algorithm to automatically segment the spots in the microarray images, and compare our results with those obtained by commercial software.

We use an optimization technique to accurately locate a distorted grid structure in a microarray image. By assuming that spot centers deviate smoothly from a checkerboard grid structure, we show that the process of gridding spot centers can be formulated as a constrained optimization problem. The constraint is equal to the variations of the transform parameter. We demonstrate the accuracy of our algorithm on two sets of microarray images. One set consists of some images from the Stanford Microarray Database; we compare our centers with those annotated in the Database. The other set consists of oligonucleotide images, and we compare our results with those obtained by GenePix Pro 5.0. Our experiments were performed completely automatically.

Complex-valued wavelets are normally used to measure instantaneous frequencies, while real wavelets are normally used to detect singularities. We prove that the wavelet modulus maxima with a complex-valued wavelet can detect and characterize singularities. This is an extension of the previous wavelet work of Mallat and Hwang on modulus maxima using a real wavelet. With this extension, we can simultaneously detect instantaneous frequencies and singularities from the wavelet modulus maxima of a complex-valued wavelet. Some results of singularity detection with the modulus maxima from a real wavelet and an analytic complex-valued wavelet are compared. We also demonstrate that singularity detection methods can be employed to detect the corners of a planar object.