第31卷第3期Vol 31MICROCOMPUTER APPLICATIONS壓縮感知及應(yīng)用李卓凡12閆敬文2(韓山師范學(xué)院物理與電子工程系潮州5210412汕頭大學(xué)工學(xué)院汕頭515063摘要:傳統(tǒng)的信號(hào)采樣必須遵循香農(nóng)采樣定理,產(chǎn)生的大量數(shù)據(jù)造成了存儲(chǔ)空間的浪費(fèi)。壓縮感知(CS)提出一種新的采樣理論,它能夠以遠(yuǎn)低于奈奎斯特采樣速率采樣信號(hào)。壓縮感知的基本論點(diǎn)是如果信號(hào)具有稀疏性,可投影到一個(gè)與變換基不相關(guān)的隨機(jī)矩陣并獲得遠(yuǎn)少于信號(hào)長度的測量值,再通過求解優(yōu)化問題精確重構(gòu)信號(hào)。本文詳述了壓縮感知的基本理論,壓縮感知適用的基本條件:稀疏性和非相干性測量矩陣設(shè)計(jì)要求,及重構(gòu)算法的RP準(zhǔn)則,并介紹了壓縮感知的應(yīng)用及仿真。仿真結(jié)果表明當(dāng)采樣個(gè)數(shù)大于 K x log(N/K),就能將N維信號(hào)穩(wěn)定地重建出來。關(guān)鍵詞:壓縮惑知觀測矩陣稀疏性RPTheory and Applicaton of Compressive SensingLI Zhuofan, YAN Jingwen(Physics and electronic engineering department, Hanshan Normal College, Chaozhou, 521041, China2College of Engineering, Shantou University, Shantou, Guangdong, 515063, China)Abstract: Conventional approaches to sampling signals follow Shannon principle. It take great costs on data storage. In this paper, therate. If signal or image is sparse in some orthonormal basis, signal or image can be recovered from small number of man optimization process. The structure of the signal is preserved in the measurement and the measure matrix is incoherent with theconormal basis, CS relies on two principles: sparsity and incoherence. RI Pprinciple is the precondition of designing reconstructionalgorithm. The application of CS theory are introduced and the simulation is illustrated in details. The simulation show thatcan be reconstructed stablely when the number of samples is larger than K x log(N/K)Keywords: corive sensing, measurement matrix; sparsity, RIP1引言傳統(tǒng)的信號(hào)采樣以奈奎斯特采樣定理為基礎(chǔ)。在獲取信號(hào)時(shí),為了不丟失信號(hào)的信息,采樣頻率必須大于信號(hào)中最高頻率的兩倍,才能精確重構(gòu)信號(hào)。但是隨著科技的迅速發(fā)展,高分辨率的數(shù)碼裝置的采樣產(chǎn)生了龐大的數(shù)據(jù),如何更高效地處理這些數(shù)據(jù)并最大限度地節(jié)省存儲(chǔ)和傳輸?shù)某杀臼且淮箅y題。實(shí)際上采樣得到的大部分?jǐn)?shù)據(jù)是不重要的,在信號(hào)或圖像的處理過程中,只保留了某些重要的數(shù)據(jù),舍棄了大量的剩余數(shù)據(jù),重構(gòu)后的信號(hào)或圖像并不會(huì)引起視覺上的差異。于是科學(xué)家們提出一個(gè)構(gòu)想,既然采集到的數(shù)據(jù)大部分都是不重要的,可以被丟棄,能否直接地采集那部分重要的、最后沒有被丟棄數(shù)據(jù),并且能夠精確地中國煤化工本文于200-12-14收到。CNMHG基金項(xiàng)目:國家自然科學(xué)基金(項(xiàng)目批準(zhǔn)號(hào):40971206)。3期李卓凡等:壓縮感知及應(yīng)用重構(gòu)原始信號(hào)或圖像。在2004年,由 Donoho等人提出了壓縮感知( compressed sensing,簡稱CS)理論1。壓縮感知理論表示:如果信號(hào)通過某種變換(如傅立葉變換,小波變換等)后,是可稀疏表示或可壓縮的,則可設(shè)計(jì)一個(gè)與變換基不相關(guān)的測量矩陣測量信號(hào),得到的測量值通過求解優(yōu)化問題,可實(shí)現(xiàn)信號(hào)的精確或近似重構(gòu)。測量后,信號(hào)∫由N維減少到M維(M<>K);或者α經(jīng)排序后按指數(shù)級衰減并趨近于零,可認(rèn)為信號(hào)是稀疏的。信號(hào)的可稀疏表示是壓縮感知的先驗(yàn)條件。在已知信號(hào)是可壓縮的前提下,壓縮感知過程可分為二步(1)設(shè)計(jì)一個(gè)與變換基不相關(guān)的MxN(M<