In signals and images processing to describe our data, we need a model that is important when implementing algorithms. The sparse land model is a powerful model for describing signals based on their sparseness and redundancy. According to this model, the signals and images can be depicted on non-orthogonal basis of a defined vector space, so that the obtained coefficients are sparse, i.e., the number of nonzero coefficients is few. The sparse representation is the same in terms of obtaining the corresponding coefficients of the signal based on specific bases with the conventional transforms such as Fourier transform, but unlike other transforms, the bases of sparse coding are not necessarily perpendicular to each other, and also the number of bases is not equal to the number of signal samples, in other words the defined vector space, known as the dictionary, has redundancy leading to sparse representation. There are various solving methods and algorithms such as matching pursuit and basis pursuit to obtain sparse representation of data that we will discuss in the following chapters. Also choosing appropriate dictionary is very important because sparse representation results from dictionary’s redundancy. Different types of dictionaries and algorithms for learning it such as K-SVD are also discussed in the following chapters.
sparse coding is used in image processing such as denoising, compression, image resizing and inpainting. Image inpainting refers to filling-in missing pixels in known locations in the image. The relationship between this issue and the sparse representation is that due to the redundancy in image information, the distorted parts can be reconstructed by obtaining sparse representation of existing pixels in the image. In these problems we can estimate the missing values with respect to other image information and reconstruct the image. In this project, we concentrate on image inpainting and its implementation using sparse coding.