Clutter Mitigation in Echocardiography Using Sparse Signal Separation

Algorithm 2

K-SVD.

Task. Train a dictionary to sparsely represent the data by approximating the solution

to problem (6).

Input Parameters. Input parameters include a matrix containing the signals and the

maximum sparsity of the solution .

Initialization. Initialization is as follows:

 (i) Initialize .

 (ii) Initialize by either using randomly chosen examples from or using random entries.

 (iii) Normalize the columns of .

Main Iteration. Increment by 1 and apply the following:

 (i) Sparse coding: obtain the sparse representations of each signal . Use OMP to

    approximate the solution of

                subject to  .

    These form the matrix .

 (ii) Dictionary update: use the following steps to update the columns of the dictionary and obtain :

     repeat for .

   (a) Define the group of samples that use the atom :

            .

   (b) Compute the residual matrix , where stands for the th row of .

   (c) Restrict by choosing only the columns corresponding to , and obtain .

   (d) Apply SVD decomposition . Update the dictionary atom and the representations .

 (iii) Stopping rule: if the change in is small enough, stop.

Output. The desired result is the dictionary and the sparse representations of the signals in .