Blind source separation based on simultaneous learning of the sparse frame representations for multi sources from their mixtures

正如任何句子都可以由词典中的几个词构成一样，任何信号或图像都可以由词典中的几个词来表示。与词典中的大量单词相比，一个句子通常只由很少的单词组成，因此这些单词被稀疏地映射到词典中。然后将构造的句子称为字典稀疏编码。同样，使用信号或图像的词典，可以表示任何信号或图像，这也可以称为稀疏编码。通常情况下，词典中的单词多于单词的长度，即词典过于完整，词典具有框架结构(一个数学概念)。本研究的目的是寻找更有效的方法来寻找这种稀疏表示，并将其应用于盲源分离，利用这种稀疏表示的盲源分离通常包括重复的两个步骤，一旦它被赋予一个任意的…估计词典的更多y初始化。在第一步中，对于给定的词典，进行该词典的稀疏表示。在第二步中，部分学习字典并修改相应的稀疏表示，而保留稀疏表示的其他部分。在这两个步骤中，词典学习更为重要，有效的学习方法仍然很少。为此，我们提出了一种称为自适应非正交稀疏变换的方法。在该方法中，我们将帧与稀疏矩阵的乘积作为源信号估计。虽然可能有许多可能的解决方案，但我们选择最稀疏的一个作为我们的结果。该方法的一个特点是将词典中的词按能量排序，而不是像通常的词典那样随机排序，词典学习和信源估计的收敛速度很慢，计算量也很大。为了解决这些问题，我们还提出了一种通过调用非负矩阵分解(NMF)来同时估计词典和源的方法。在应用中存在许多非负信号，如图像。然而，通常情况下，NMF的结果并不是唯一的，也不是所有的结果都是稀疏的。为了解决这个问题，我们提出了一种稀疏NMF方法，通过约束从非唯一解中选择一个稀疏解。我们提出了一种衡量源信号稀疏性的度量，并以其最小化为约束条件。作为一种替代方法，我们还建议使用对词典的约束，而不是对源的约束，因为源通常很长，对它们的约束将耗费计算。我们发现最大化词典中单词所占空间是一个很好的约束，评估结果表明我们的方法是有效的。然后，我们将它们应用于盲谱分解、图像的盲源分离或去噪、波束形成和到达方向估计。较少

Just as that any sentence can be constructed by several words in a dictionary, any signal or image can be either represented by several "words" in a "dictionary". Comparing with the large number of words in the dictionary, a sentence is usually be constructed by only very few words, so that these words are mapped into the dictionary sparsely. Then constructed sentence may be called as sparse coding with the dictionary. Similarly, with a dictionary for signal or image, one can represent any signal or image, and this can also be termed as sparse coding. Usually, there more words in the dictionary than the length of the words, i.e., the dictionary is over-complete, the dictionary has a structure of frame (a mathematical concept). The motivation of this research is to find more effective methods for finding this sparse representation, and then apply them to blind source separation (BSS).BSS by using the sparse representation usually includes repeated two steps, once it is given an arbitrar … More y initialization of the estimated dictionary. In the first step, for a given dictionary the sparse representation by the dictionary is conducted. In the second step, the dictionary is learned in part and the corresponding sparse representation is modified, while the other parts of sparse representation are kept. In these two steps, the dictionary learning is more important and there still very few effective methods for it. For this purpose, we worked out a method that is termed as adaptive non-orthogonal sparsifying transform. In this method, we take the multiplication of the frame and a sparse matrix as the source signal estimation. Though there may be many possible solutions, we choose the sparsest one as our result. As a feature of the method, the words in the dictionary are ordered by their energy, rather than randomly ordered as in the usual dictionary.In the above method, the dictionary learning and source estimation may converge very slowly and the computation is also very consuming. For solving these problems, we also worked out a method in which the dictionary and the sources are simultaneously estimated, by invoking the nonnegative matrix factorization (NMF). There are many nonnegative signals, such as image, in applications. However, usually, the result of NMF is not unique and not all of them are sparse. For solving this problem, we worked out a sparse NMF method, in which we select a sparse solution from the non unique solutions by a constraint. We propose a measure for measuring the sparsity of source signals and use its minimization as the constraint. As an alternative method, we also proposed to use a constraint on the dictionary, rather than on the sources, since the sources are usually very long and a constraint on them will be computation consuming. We found that the maximization of space spanned by the words in the dictionary is a good constraint.Evaluations showed that our methods are efficient. Then we applied them to blind spectral unmixing, BSS or denoising of images, beamforming and direction of arrival estimation. Less