Sparsity Representation of a Signal Using the DCT Matrix

1. What is sparsity in signal processing?
2. What is meant by sparse representation?
3. Why and when do we use sparse representation?

What is sparsity in signal processing?

A signal is considered sparse if most of its information is contained within a few non-zero samples. Consequently, a signal reconstruction algorithm has to find a sparse vector that best represents the measured signal. Many algorithms to solve this problem are based on l1-norm optimization.

What is meant by sparse representation?

Sparse representations intend to represent signals with as few as possible significant coefficients. This is important for many applications, like for instance compression. When using wavelets it is frequently noticed that a great compression rate can be obtained, with almost unnoticeable loss of information.

Why and when do we use sparse representation?

Sparse representation attracts great attention as it can significantly save computing resources and find the characteristics of data in a low-dimensional space. Thus, it can be widely applied in engineering fields such as dictionary learning, signal reconstruction, image clustering, feature selection, and extraction.