SHAH: SHape-Adaptive Haar wavelets for image processing

We propose the SHAH (SHape-Adaptive Haar) transform for images, which results in an orthonormal, adaptive decomposition of the image into Haar-wavelet-like components, arranged hierarchically according to decreasing importance, whose shapes reflect the features present in the image. The decomposition is as sparse as it can be for piecewise-constant images. It is performed via an stepwise bottom-up algorithm with quadratic computational complexity; however, nearly-linear variants also exist. SHAH is rapidly invertible. We show how to use SHAH for image denoising. Having performed the SHAH transform, the coefficients are hard- or soft-thresholded, and the inverse transform taken. The SHAH image denoising algorithm compares favourably to the state of the art for piecewise-constant images. A clear asset of the methodology is its very general scope: it can be used with any images or more generally with any data that can be represented as graphs or networks

Magnetic resonance image reconstruction using trained geometric directions in 2D redundant wavelets domain and non-convex optimization

Reducing scanning time is significantly important for MRI. Compressed sensing has shown promising results by undersampling the k-space data to speed up imaging. Sparsity of an image plays an important role in compressed sensing MRI to reduce the image artifacts. Recently, the method of patch-based directional wavelets (PBDW) which trains geometric directions from undersampled data has been proposed. It has better performance in preserving image edges than conventional sparsifying transforms. However, obvious artifacts are presented in the smooth region when the data are highly undersampled. In addition, the original PBDW-based method does not hold obvious improvement for radial and fully 2D random sampling patterns. In this paper, the PBDW-based MRI reconstruction is improved from two aspects: 1) An efficient non-convex minimization algorithm is modified to enhance image quality; 2) PBDW are extended into shift-invariant discrete wavelet domain to enhance the ability of transform on sparsifying piecewise smooth image features. Numerical simulation results on vivo magnetic resonance images demonstrate that the proposed method outperforms the original PBDW in terms of removing artifacts and preserving edges

Joint sparse learning with nonlocal and local image priors for image error concealment

Joint sparse representation (JSR) model has recently emerged as a powerful technique with wide variety of applications. In this paper, the JSR model is extended to error concealment (EC) application, being effective to recover the original image from its corrupted version. This model is based on jointly learning a dictionary pair and two mapping matrices that are trained offline from external training images. Given the trained dictionaries and mappings, the restoration is done by transferring the recovery problem into the sparse representation domain with respect to the trained dictionaries, which is further transformed into a common space using the respective mapping matrices. Then, the reconstructed image is obtained by back projection into the spatial domain. In order to improve the accuracy and stability of the proposed JSR-based EC algorithm and avoid unexpected artifacts, the local and non-local priors are seamlessly integrated into the JSR model. The non-local prior is based on the self-similarity within natural images and helps to find an accurate sparse representation by taking a weighted average of similar areas throughout the image. The local prior is based on learning the local structural regularity of the natural images and helps to regularize the sparse representation, exploiting the strong correlation in the small local areas within the image. Compared with the state-of-the-art EC algorithms, the results show that the proposed method has better reconstruction performance in terms of objective and subjective evaluations

Mathematical Approaches for Image Enhancement Problems

This thesis develops novel techniques that can solve some image enhancement problems using theoretically and technically proven and very useful mathematical tools to image processing such as wavelet transforms, partial differential equations, and variational models. Three subtopics are mainly covered. First, color image denoising framework is introduced to achieve high quality denoising results by considering correlations between color components while existing denoising approaches can be plugged in flexibly. Second, a new and efficient framework for image contrast and color enhancement in the compressed wavelet domain is proposed. The proposed approach is capable of enhancing both global and local contrast and brightness as well as preserving color consistency. The framework does not require inverse transform for image enhancement since linear scale factors are directly applied to both scaling and wavelet coefficients in the compressed domain, which results in high computational efficiency. Also contaminated noise in the image can be efficiently reduced by introducing wavelet shrinkage terms adaptively in different scales. The proposed method is able to enhance a wavelet-coded image computationally efficiently with high image quality and less noise or other artifact. The experimental results show that the proposed method produces encouraging results both visually and numerically compared to some existing approaches. Finally, image inpainting problem is discussed. Literature review, psychological analysis, and challenges on image inpainting problem and related topics are described. An inpainting algorithm using energy minimization and texture mapping is proposed. Mumford-Shah energy minimization model detects and preserves edges in the inpainting domain by detecting both the main structure and the detailed edges. This approach utilizes faster hierarchical level set method and guarantees convergence independent of initial conditions. The estimated segmentation results in the inpainting domain are stored in segmentation map, which is referred by a texture mapping algorithm for filling textured regions. We also propose an inpainting algorithm using wavelet transform that can expect better global structure estimation of the unknown region in addition to shape and texture properties since wavelet transforms have been used for various image analysis problems due to its nice multi-resolution properties and decoupling characteristics

Tungsten tip and atomic site tomography

Atomic electron tomography aims to precisely locate individual atoms of a nanoparticle in three-dimensional space. In this work, a tomography method based on tungsten tips is developed to allow images to be taken over a full angular range by placing a nanoparticle on the apex of an etched tungsten tip. There is no interference of signal from supporting materials with the suspended nanoparticle. A new reconstruction algorithm, atomic site tomography, is developed using the principle of regularisation in multiple linear regression. This algorithm is specifically designed for identifying the precise locations of individual atoms in three-dimensional space, and the algorithm is validated by an experimental dataset. A gold nanoparticle dataset is successfully obtained by tungsten tip tomography, and the dataset is processed to remove scanning artefacts. Selected region of the gold nanoparticle dataset is used to demonstrate the new reconstruction algorithm and the whole gold nanoparticle is then reconstructed. A tuning fork atomic force microscope is developed to provide a more flexible method to prepare samples for tungsten tip tomography and its progress is reported. This work contributes to the field of atomic electron tomography by improving the experimental techniques for acquiring high-quality tomography dataset and proposing a new reconstruction algorithm which aims at locating individual atoms of nanoparticles precisely