【正文】 trainingdenoising algorithm bees fused into one iterative procedure that prises of steps of denoising of the image, followed by an update of the dictionary. This is described in Section III in detail. In Section IV, we show some experimental results that demonstrate the effectiveness of this algorithm.II. FROM LOCAL TO GLOBAL BAYESIAN RECONSTRUCTIONIn this section, we start the presentation of the proposed denoising algorithm by first introducing how sparsity and redundancy are brought to use. We do that via the introduction of the Sparseland model. Once this is set, we will discuss how local treatment on image patches turns into a global prior in a Bayesian reconstruction framework.A. Sparseland Model for Image PatchesWe consider image patches of size pixels, ordered lexicographically as column vectors . For the construction of the Sparseland model, we need to define a dictionary (matrix) of size (with , implying that it is redundant). At the moment, we shall assume that this matrix is known and fixed. Put loosely, the proposed model suggests that every image patch, , could be represented sparsely over this dictionary, ., the solution of subject to (2)is indeed very sparse, . The notation stands for the count of the nonzero entries in . The basic idea here is that every signal instance from the family we consider can be represented as a linear bination of few columns (atoms)from the redundant dictionary .This model should be made more precise by replacing the rough constraint with a clear requirement to allow a bounded representation error, . Also, one needs to define how deep is the required sparsity, adding a requirement of the form , that states that the sparse representation uses no more than atoms from the dictionary for every image patch instance. Alternatively, a probabilistic characterizationcan be given, defining the probability to obtain a representation with nonzeros as a decaying function of some sort. Considering the simpler option between the two, with the triplet in place, our model is well defined.Now assume that indeed belongs to the Sparseland signals. Consider a noisy version of it,y , contaminated by an additive zeromean white Gaussian noise with standard deviation . The MAP estimator for denoising this image patch is built by solving subject to (3)where is dictated by . The denoised image is, thus, given by [22], [41], [42]. Notice that the above optimization task can be changed to be (4)so that the constraint bees a penalty. For a proper choice of, the two problems are equivalent. We will use this alternative terminology from now on, as it makes the presentation of later parts simpler to follow.While this problem is, in general, very hard to solve, the matching and the basis pursuit algorithms can be used quite effectively [20]–[22] to get an approximated solution. Recentwork established that those approximation techniques can be quite accurate if the solution is sparse enough to begin with[41], [42]. In this work, we will make use mainly of the orthonormal matching pursuit (OMP) because of its simplicity [21]and efficiency.B. From Local Analysis to a Global PriorIf we want to handle a larger image X of size , and we are still interested in using the above described model, one option is to redefine the model with a larger dictionary. Indeed, when using this model with a dictionary emerging from the contourlet or curvelet transforms, such scaling is simple and natural [26].However, when we insist on using a specific fixed and small size dictionary , this option no longer exists. Thus,a natural question arises concerning the use of such a small dictionary in the first place. Two reasons e to mind: 1) when training takes place (as we will show in the next section), only small dictionaries can be posed。5 結(jié)論與展望本文我們系統(tǒng)地研究和學習了基于基元組的稀疏線性表達的方法及其在圖像去噪中的應(yīng)用。①給定，；②給定，初始化：用中值濾波對噪聲圖像做去噪處理得到初始去噪圖像，采用高斯函數(shù)，=超完備DCT基元組。對問題（1）， （35）與經(jīng)典模型的區(qū)別在于在第一個懲罰項中加入了權(quán)重向量，將上式寫為， （36）問題的求解同經(jīng)典稀疏表達模型類似，我們?nèi)匀徊捎谜黄ヅ渥粉櫍∣MP）對稀疏表達系數(shù)求解?？偠灾?，就是我們盡可能只使用圖像中那些受噪聲影響較小的點學習稀疏表達系數(shù)。如果應(yīng)用范數(shù)測度描述時會顯得非常不魯棒，使得學習到的基元表達系數(shù)受到椒鹽噪聲的嚴重影響，影響去噪精度。，初始化：令，=超完備DCT基元組。回到（26），我們可以將看做未知的，并定義我們的模型為 （211） 根據(jù)先前構(gòu)建的算法，我們可以初始化基元組和整體去噪圖像，和先前的處理一樣設(shè)為DCT基元組。這個過程即是我們通過稀疏表達來迭代去除噪聲[13][15]。我們從初始化開始，尋找最優(yōu)值。作為約束，此懲罰項,這反映了和之間的關(guān)系。于是去噪圖像就可以由給出。如下： . (22)上式中使用范數(shù)對線性組合系數(shù)的稀疏性進行約束。該類算法和模型的基本思想是：首先將原始圖像劃分為一個個小的圖像塊，然后將原始圖像表達為局部的基元線性組合，并約束這個線性組合系數(shù)的稀疏性，從而建立解決去噪問題的能量函數(shù)，在極小化過程中將通過OMP和KSVD算法優(yōu)化該能量函數(shù)。因此我們在研究改進工作時，考慮引進對圖像像素點的噪聲可能性的權(quán)重函數(shù)，并建立帶權(quán)的稀疏表達模型，減少噪聲點對稀疏表達模型的影響。從后文中我們可以看到使用OMP算法可以在每個局部塊上求解出稀疏表達系數(shù)。之所以用到冗余表達是因為我們希望在處理圖像去噪問題過程中能保持轉(zhuǎn)換不變性，與此同時我們引入匹配追蹤技術(shù)[8]可以很方便地優(yōu)化問題求解過程中的稀疏表達系數(shù)[9][12]。本文主要研究兩類噪聲的去噪問題，即：①高斯噪聲，所謂高斯噪聲是指它的概率密度函數(shù)服從高斯分布（即正態(tài)分布）的一類噪聲；②椒鹽噪聲，在該噪聲影響下，圖像像素點會變?yōu)?個極值灰度（例如0，255），而圖像中的每個像素點以一定的概率受到該噪聲的影響，因此它表現(xiàn)為圖象某些點特別暗或特別亮，而其他象素點不受到影響，類似我們的胡椒粉和晶體鹽的亮度的感覺，所以叫椒鹽噪聲。OMP。在實現(xiàn)中，我們可以用離散余弦變換（DCT）構(gòu)造其中的基元組，也可以自適應(yīng)的學習該基元組。本文主要研究基于稀疏表達的高斯噪聲和椒鹽噪聲去噪模型與算法。我們發(fā)現(xiàn)，應(yīng)用經(jīng)典的稀疏表達模型會在處理去除椒鹽噪聲圖像中失效，因此我們提出一種新的基于稀疏性的椒鹽噪聲圖像去噪模型。上個世紀60年代中期，隨著計算機科學的發(fā)展和計算機的普及，圖像處理得到廣泛的應(yīng)用。 過去對于圖像去噪問題的研究有著眾多不同的看法和觀點[1][7]。此模型是通過定義關(guān)于的后驗概率分布并進行優(yōu)化而引出的能量模型。我們實現(xiàn)上述模型的數(shù)值解法，并應(yīng)用于自然圖像的高斯噪聲去噪問題。第4章——在這章里我們將展示一些實驗結(jié)果以表明我們建立的模型及算法的有效性，在這一章節(jié)中我們將看到利用建立起的經(jīng)典模型對高斯噪聲去噪有著相當不錯的效果，并且改進后的模型在處理椒鹽噪聲去噪時比經(jīng)典模型有更好的表現(xiàn)。再將其推廣到較大的圖像上。對稀疏表達系數(shù)的稀疏強度我們需要作出定義，令，這表明用稀疏線性組合來表達圖像塊最多只使用了基元組中的個基元。假如我們要處理的未知圖像大小為，仍然可以從圖像中取出圖像塊。對于基元組，采用離散余弦變換（DCT）確實是一個相當不錯的選擇。因此，這個階段又稱稀疏編碼階段，隨著滑動的窗口在每個塊上同時運算。因此，可以繼續(xù)使用OMP獲得近似最優(yōu)的稀疏表達系數(shù)。 輸出圖像應(yīng)用（29）式可以得到。本章將研究如何對經(jīng)典模型進行改進，使得新的模型對椒鹽噪聲有較好的去噪效果。整個模型優(yōu)化形式如下：（1）給定，這里初始為原來的噪聲圖像，假設(shè)已知，設(shè)為DCT基元組 （32）（2）給定。我們可以看出式（33）中第一個懲罰項和第二個懲罰項相互競爭。實驗過程中我們使用標準的數(shù)據(jù)測試：所有要處理的圖片大小為512512，DCT基元組大小為64256，用來處理圖像塊的大小為88像素，高斯噪聲模型中我們設(shè)，椒鹽噪聲模型中噪聲強度統(tǒng)一為P=。實現(xiàn)算法方面，我們在稀疏編碼階段常采用正交匹配追蹤（OMP）方法，應(yīng)用KSVD算法對基元組迭