WARP: Wavelets with adaptive recursive partitioning for multi-dimensional data

Effective identification of asymmetric and local features in images and other data observed on multi-dimensional grids plays a critical role in a wide range of applications including biomedical and natural image processing. Moreover, the ever increasing amount of image data, in terms of both the resolution per image and the number of images processed per application, requires algorithms and methods for such applications to be computationally efficient. We develop a new probabilistic framework for multi-dimensional data to overcome these challenges through incorporating data adaptivity into discrete wavelet transforms, thereby allowing them to adapt to the geometric structure of the data while maintaining the linear computational scalability. By exploiting a connection between the local directionality of wavelet transforms and recursive dyadic partitioning on the grid points of the observation, we obtain the desired adaptivity through adding to the traditional Bayesian wavelet regression framework an additional layer of Bayesian modeling on the space of recursive partitions over the grid points. We derive the corresponding inference recipe in the form of a recursive representation of the exact posterior, and develop a class of efficient recursive message passing algorithms for achieving exact Bayesian inference with a computational complexity linear in the resolution and sample size of the images. While our framework is applicable to a range of problems including multi-dimensional signal processing, compression, and structural learning, we illustrate its work and evaluate its performance in the context of 2D and 3D image reconstruction using real images from the ImageNet database. We also apply the framework to analyze a data set from retinal optical coherence tomography

Image Restoration Methods for Retinal Images: Denoising and Interpolation

Retinal imaging provides an opportunity to detect pathological and natural age-related physiological changes in the interior of the eye. Diagnosis of retinal abnormality requires an image that is sharp, clear and free of noise and artifacts. However, to prevent tissue damage, retinal imaging instruments use low illumination radiation, hence, the signal-to-noise ratio (SNR) is reduced which means the total noise power is increased. Furthermore, noise is inherent in some imaging techniques. For example, in Optical Coherence Tomography (OCT) speckle noise is produced due to the coherence between the unwanted backscattered light. Improving OCT image quality by reducing speckle noise increases the accuracy of analyses and hence the diagnostic sensitivity. However, the challenge is to preserve image features while reducing speckle noise. There is a clear trade-off between image feature preservation and speckle noise reduction in OCT. Averaging multiple OCT images taken from a unique position provides a high SNR image, but it drastically increases the scanning time. In this thesis, we develop a multi-frame image denoising method for Spectral Domain OCT (SD-OCT) images extracted from a very close locations of a SD-OCT volume. The proposed denoising method was tested using two dictionaries: nonlinear (NL) and KSVD-based adaptive dictionary. The NL dictionary was constructed by adding phases, polynomial, exponential and boxcar functions to the conventional Discrete Cosine Transform (DCT) dictionary. The proposed denoising method denoises nearby frames of SD-OCT volume using a sparse representation method and combines them by selecting median intensity pixels from the denoised nearby frames. The result showed that both dictionaries reduced the speckle noise from the OCT images; however, the adaptive dictionary showed slightly better results at the cost of a higher computational complexity. The NL dictionary was also used for fundus and OCT image reconstruction. The performance of the NL dictionary was always better than that of other analytical-based dictionaries, such as DCT and Haar. The adaptive dictionary involves a lengthy dictionary learning process, and therefore cannot be used in real situations. We dealt this problem by utilizing a low-rank approximation. In this approach SD-OCT frames were divided into a group of noisy matrices that consist of non-local similar patches. A noise-free patch matrix was obtained from a noisy patch matrix utilizing a low-rank approximation. The noise-free patches from nearby frames were averaged to enhance the denoising. The denoised image obtained from the proposed approach was better than those obtained by several state-of-the-art methods. The proposed approach was extended to jointly denoise and interpolate SD-OCT image. The results show that joint denoising and interpolation method outperforms several existing state-of-the-art denoising methods plus bicubic interpolation.4 month

BAYESIAN ENSEMBLE LEARNING FOR MEDICAL IMAGE DENOISING

Medical images are often affected by random noise because of both image acquisition from the medical modalities and image transmission from modalities to workspace in the main computer. Medical image denoising removes noise from the CT or MR images and it is an essential step that makes diagnosing more efficient. Many denoising algorithms have been introduced such as Non-local Means, Fields of Experts, and BM3D. In this thesis, we implement the Bayesian ensemble learning for not only natural image denoising but also medical image denoising. The Bayesian ensemble models are Non-local Means and Fields of Experts, the very successful recent algorithms. The Non-local Means presumes that the image contains an extensive amount of self-similarity. The approach of the Fields of Experts model extends traditional Markov Random Field model by learning potential functions over extended pixel neighborhoods. The two models are implemented, and image denoising is performed on both natural images and MR images. For MR images, we used two noise distributions, Gaussian and Rician. The experimental results obtained are used to compare with the single algorithm, and discuss the ensemble learning and their approaches