稀疏性在最近的圖像恢復技術發展中起到了重要作用。在這個碩士研究中，我們專注於兩種通過信號稀疏性假設相聯繫的圖像恢復問題。具體來講，在第一個圖像恢復問題中，信號本身在某些變換域是稀疏的，例如小波變換。在本研究的第二部分，信號並非傳統意義上的稀疏，但它可以用很少的幾個參數來表示--亦即信號具有稀疏的表示。我們希望通過講述一個「雙城記」，聯繫起這兩個稀疏圖像重建問題。 / 在第二章中，我們提出了一種創新的算法框架，用於解決信號稀疏假設下的圖像恢復問題。重建圖像的目標函數，由一個數據保真項和`1正則項組成。然而，我們不是直接估計重建的圖像，而是專注於如何獲得重建的這個過程。我們的策略是將這個重建過程表示成基本閾值函數的線性組合（LET）：這些線性係數可以通過最小化目標函數解得。然後，可以更新閾值函數并迭代這個過程（i-LET）。這種線性參數化的主要優點是可以大幅降低問題的規模-每次我們只需解決一個線性係數維度大小的優化問題（通常小於十），而不是整個圖像大小的問題。如果閾值函滿足一定的條件，迭代LET算法可以保證全局的收斂性。多個測試圖像在不同噪音水平和不同卷積核類型的測試清楚地表明，我們提出的框架在所需運算時間和迭代循環次數方面，通常超越當今最好水平。 / 在第三章中，我們擴展了有限創新率採樣框架至某一種特定二維曲線。我們用掩模函數的解來間接定義這個二維曲線。這裡，掩模函數可以表示為有限數目的正弦信號加權求和。因此，從這個角度講，我們定義的二維曲線具有「有限創新率」（FRI）。由於與定義曲線相關聯的指示器圖像沒有帶寬限制，因而根據經典香農採樣定理，不能在有限數量的採樣基礎上獲得完全重建。然而，我們證明，仍然可以設計一個針對指示器圖像採樣的框架，實現完美重構。此外，對於這一方法的空間域解釋，使我們能夠拓展嚴格的FRI曲線模型用於描述自然圖像的邊緣，可以在各種圖像處理的問題中保持圖像的邊緣。我們用一個潛在的在圖像上採樣中的應用作為示例。 / Sparsity has played an important role in recent developments of various image restoration techniques. In this MPhil study, we focus on two different types of image restoration problems, which are related by the sparsity assumptions. Specifically, in the first image restoration problem, the signal (i.e. the restored image) itself is sparse in some transformation domain, e.g. wavelet. While in the second part of this study, the signal is not sparse in the traditional sense but that it can be parametrized with a few parameters hence having a sparse representation. Our goal is to tell a "tale of two cities" and to show the connections between the two sparse image restoration problems in this thesis. / In Chapter 2, we proposed a novel algorithmic framework to solve image restoration problems under sparsity assumptions. As usual, the reconstructed image is the minimum of an objective functional that consists of a data fidelity term and an ℓ₁ regularization. However, instead of estimating the reconstructed image that minimizes the objective functional directly, we focus on the restoration process that maps the degraded measurements to the reconstruction. Our idea amounts to parameterizing the process as a linear combination of few elementary thresholding functions (LET) and solve for the linear weighting coefficients by minimizing the objective functional. It is then possible to update the thresholding functions and to iterate this process (i-LET). The key advantage of such a linear parametrization is that the problem size reduces dramatically--each time we only need to solve an optimization problem over the dimension of the linear coefficients (typically less than 10) instead of the whole image dimensio . With the elementary thresholding functions satisfying certain constraints, global convergence of the iterated LET algorithm is guaranteed. Experiments on several test images over a wide range of noise levels and different types of convolution kernels clearly indicate that the proposed framework usually outperform state-of-theart algorithms in terms of both CPU time and number of iterations. / In Chapter 3, we extended the sampling framework for signals with finite rate of innovation to a specific class of two-dimensional curves, which are defined implicitly as the roots of a mask function. Here the mask function has a parametric representation as weighted summation of a finite number of sinusoids, and therefore, has finite rate of innovation [1]. The associated indicator image of the defined curve is not bandlimited and cannot be perfectly reconstructed based on the classical Shannon's sampling theorem. Yet, we show that it is possible to devise a sampling scheme and have a perfect reconstruction from finite number of (noiseless) samples of the indicator image with the annihilating filter method (also known as Prony's method). Robust reconstruction algorithms with noisy samples are also developed. Furthermore, the new spatial domain interpretation of the annihilating filter enables us to generalize the exact FRI curve model to characterize edges of a natural image. We can impose the annihilation constraint to preserve edges in various image processing problems. We exemplified the effectiveness of the annihilation constraint with a potential application in image up-sampling. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Pan, Hanjie. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 69-74). / Abstracts also in Chinese. / Acknowledgments --- p.iii / Abstract --- p.vii / Contents --- p.xii / List of Figures --- p.xv / List of Tables --- p.xvii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Sampling Sparse Signals --- p.1 / Chapter 1.2 --- Thesis Organizations and Contributions --- p.3 / Chapter 2 --- An Iterated Linear Expansion of Thresholds for ℓ₁-based Image Restoration --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.1.1 --- Problem Description --- p.5 / Chapter 2.1.2 --- Approaches to Solve the Problem --- p.6 / Chapter 2.1.3 --- Proposed Approach --- p.8 / Chapter 2.1.4 --- Organization of the Chapter --- p.9 / Chapter 2.2 --- Basic Ingredients --- p.9 / Chapter 2.2.1 --- Iterative Reweighted Least Square Methods --- p.9 / Chapter 2.2.2 --- Linear Expansion of Thresholds (LET) --- p.11 / Chapter 2.3 --- Iterative LET Restoration --- p.15 / Chapter 2.3.1 --- Selection of i-LET Bases --- p.15 / Chapter 2.3.2 --- Convergence of the i-LET Scheme --- p.16 / Chapter 2.3.3 --- Examples of i-LET Bases --- p.18 / Chapter 2.4 --- Experimental Results --- p.23 / Chapter 2.4.1 --- Deconvolution with Decimated Wavelet Transform --- p.24 / Chapter 2.4.2 --- Deconvolution with Redundant Wavelet Transform --- p.28 / Chapter 2.4.3 --- Algorithm Complexity Analysis --- p.29 / Chapter 2.4.4 --- Choice of Regularization Weight λ --- p.30 / Chapter 2.4.5 --- Deconvolution with Cycle Spinnings --- p.30 / Chapter 2.5 --- Summary --- p.31 / Chapter 3 --- Sampling Curves with Finite Rate of Innovation --- p.33 / Chapter 3.1 --- Introduction --- p.33 / Chapter 3.2 --- Two-dimensional Curves with Finite Rate of Innovation --- p.34 / Chapter 3.2.1 --- FRI Curves --- p.34 / Chapter 3.2.2 --- Interior Indicator Image --- p.35 / Chapter 3.2.3 --- Acquisition of Indicator Image Samples --- p.36 / Chapter 3.3 --- Reconstruction of the Annihilable Curves --- p.37 / Chapter 3.3.1 --- Annihilating Filter Method --- p.37 / Chapter 3.3.2 --- Relate Fourier Transform with Spatial Domain Samples --- p.39 / Chapter 3.3.3 --- Reconstruction of Annihilation Coe cients --- p.39 / Chapter 3.3.4 --- Reconstruction with Model Mismatch --- p.42 / Chapter 3.3.5 --- Retrieval of the Annihilable Curve Amplitudes --- p.46 / Chapter 3.4 --- Dealing with Non-ideal Low-pass Filtered Samples --- p.48 / Chapter 3.5 --- Generalization of the FRI Framework for Natural Images --- p.49 / Chapter 3.5.1 --- Spatial Domain Interpretation of the Annihilation Equation --- p.50 / Chapter 3.5.2 --- Annihilable Curve Approximation of Image Edges --- p.51 / Chapter 3.5.3 --- Up-sampling with Annihilation Constraint --- p.53 / Chapter 3.6 --- Conclusion --- p.57 / Chapter 4 --- Conclusions --- p.59 / Chapter 4.1 --- Thesis Summary --- p.59 / Chapter 4.2 --- Perspectives --- p.60 / Chapter A --- Proofs and Derivations --- p.61 / Chapter A.1 --- Proof of Lemma 3 --- p.61 / Chapter A.2 --- Proof of Theorem 2 --- p.62 / Chapter A.3 --- Efficient Implementation of IRLS Inner Loop with Matlab --- p.63 / Chapter A.4 --- Derivations of the Sampling Formula (3.7) --- p.64 / Chapter A.5 --- Correspondence between the Spatial and Fourier Domain Samples --- p.65 / Chapter A.6 --- Optimal Post-filter Applied to Non-ideal Samples --- p.66 / Bibliography --- p.69
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328503 |
| Date | January 2013 |
| Contributors | Pan, Hanjie., Chinese University of Hong Kong Graduate School. Division of Electronic Engineering. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (xvii, 74 leaves) : ill. |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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