Ph.D Thesis

Ph.D StudentRubinstein Ron
SubjectAnalysis and Synthesis Sparse Modeling Methods in Image
DepartmentDepartment of Computer Science
Supervisor PROF. Michael Elad
Full Thesis textFull thesis text - English Version


Signal models are a cornerstone of contemporary signal and image processing methodology. Of these models, analysis and synthesis sparse representation models have been particularly successful in a wide range of applications. Both models take a decompositional approach, and describe signals in terms of an underlying set, or dictionary, of elementary signals known as atoms. Analysis models describe signals in terms of their inner products with the atoms, whereas synthesis models describe signals as linear combinations of atoms. The driving force behind both is sparsity - the rapid decay of the representation coefficients over the dictionary. The two models have been found effective in a wide range of applications, including denoising, demosaicing, compression, inpainting, upscaling, compressive sensing, and more.

This thesis studies several aspects of the analysis and synthesis modeling paradigms. We begin with the question of the relation between the two models, and show, through geometrical reasoning, that contrary to the mathematical similarity, the two approaches are in fact generally distinct, with a wide gap separating the two. These results ignite a renewed interest in the analysis formulation, and provide several insights about this model.

In the main part of the thesis we focus on the core component of these models - the dictionary. We describe the two main disciplines for designing such dictionaries - harmonic analysis and machine learning - and discuss the recent trend of converging the two through parametric dictionaries. We develop a specific parametric dictionary which we name the sparse dictionary, and which provides a simple and expressive structure for designing adaptable and efficient dictionaries. Among the applications of this structure, we describe a unique system for compressing generic images, which encodes each input image over a specifically-trained dictionary, sent as part of the compressed stream.

In the last part of this thesis, we return to the analysis formulation and consider the problem of dictionary training for analysis models. We present two approaches to the training problem. The first trains a dictionary for a new L0 analysis model, which is largely motivated by the geometrical understanding of the analysis structure. The second method trains a pair of analysis and synthesis dictionaries for thresholding-based image recovery, and provides a simple and effective framework for developing image recovery processes. We find that the analysis framework thus presents a promising new field, which is well-situated to complement or compete with the synthesis approach.