位元平面之壓縮取樣搭配貝氏估計解碼法於失真壓縮之應用

Abstract

本論文探討的問題是：當壓縮取樣點已遭受失真破壞，甚至取樣點亦嚴重不足的情況下，如何能夠有效地重建回原本的稀疏訊號。而這樣的過程涉及如何估計稀疏訊號子空間和稀疏訊號非零係數，在本研究中，我們依據最大後驗機率估計理論(MAP)，推導了一組向量化的估計函數。藉著充分利用訊號的前驗機率，使得我們提出的作法，可以利用極少、接近稀疏訊號非零係數個數的取樣點，即可進行完美重建！同時，在取樣點嚴重不足的情況下預測稀疏訊號子空間時，本方法亦比前人方法有更低的錯誤預測機率。此外，為了能更佳地重建稀疏訊號的重要資訊，我們提出了位元平面切割的方法。當這樣的技術與MAP估計函數相結合後，我們實證其可得到與泛用JPEG標準相匹敵的壓縮性能：在同樣的壓縮品質下，其壓縮率差距小於一倍。除此之外，我們亦將這個嶄新的技術應用於多重描述編碼，結果亦顯示我們設計的MAP估計方法，即使是在流失率極嚴重的摧毀式通道環境，還是能夠提供明顯良好的解碼品質。總而言之，相較前人研究成果，我們的壓縮取樣系統搭配MAP貝氏估計解碼法，不僅能提供極佳的壓縮性能，同時亦保有極為強悍的錯誤扺抗能力。

This thesis addresses the problem of reconstructing a compressively sampled sparse signal from its lossy and possibly insufficient measurements. The process involves estimations of sparsity pattern and sparse representation, for which we derived a vector estimator based on the Maximum a Posteriori Probability (MAP) rule. By making full use of signal prior knowledge, our scheme can use a measurement number close to sparsity to achieve perfect reconstruction. It also shows a much lower error probability of sparsity pattern than prior work, given insufficient measurements. To better recover the most significant part of the sparse representation, we further introduce the notion of bit-plane separation. When applied to image compression, the technique in combination with our MAP estimator shows promising results as compared to JPEG: the difference in compression ratio is seen to be within a factor of two, given the same decoded quality. More than this, we also applied such framework into the application of multiple description coding. According to the simulation results, our MAP estimator is very resilient to the loss of measurements, showing an acceptable quality while coping with a highly loss channel. In conclusion, our CS framework using the MAP estimation can provide a much better compression efficiency and stronger functionality of error-resilience, compared with other prior work.