Matching pursuit

Matching pursuit (MP) is a sparse approximation algorithm which finds the 'best matching' projections of multidimensional data onto the span of an over-complete (i.e., redundant) dictionary D {displaystyle D} . The basic idea is to approximately represent a signal f {displaystyle f} from Hilbert space H {displaystyle H} as a weighted sum of finitely many functions g γ n {displaystyle g_{gamma _{n}}} (called atoms) taken from D {displaystyle D} . An approximation with N {displaystyle N} atoms has the form Matching pursuit (MP) is a sparse approximation algorithm which finds the 'best matching' projections of multidimensional data onto the span of an over-complete (i.e., redundant) dictionary D {displaystyle D} . The basic idea is to approximately represent a signal f {displaystyle f} from Hilbert space H {displaystyle H} as a weighted sum of finitely many functions g γ n {displaystyle g_{gamma _{n}}} (called atoms) taken from D {displaystyle D} . An approximation with N {displaystyle N} atoms has the form where g γ n {displaystyle g_{gamma _{n}}} is the γ n {displaystyle gamma _{n}} th column of the matrix D {displaystyle D} and a n {displaystyle a_{n}} is the scalar weighting factor (amplitude) for the atom g γ n {displaystyle g_{gamma _{n}}} . Normally, not every atom in D {displaystyle D} will be used in this sum. Instead, matching pursuit chooses the atoms one at a time in order to maximally (greedily) reduce the approximation error. This is achieved by finding the atom that has the highest inner product with the signal (assuming the atoms are normalized), subtracting from the signal an approximation that uses only that one atom, and repeating the process until the signal is satisfactorily decomposed, i.e., the norm of the residual is small,where the residual after calculating γ N {displaystyle gamma _{N}} and a N {displaystyle a_{N}} is denoted by If R n {displaystyle R_{n}} converges quickly to zero, then only a few atoms are needed to get a good approximation to f {displaystyle f} . Such sparse representations are desirable for signal coding and compression. More precisely, the sparsity problem that matching pursuit is intended to approximately solve is where ‖ x ‖ 0 {displaystyle |x|_{0}} is the L 0 {displaystyle L_{0}} pseudo-norm (i.e. the number of nonzero elements of x {displaystyle x} ). In the previous notation, the nonzero entries of x {displaystyle x} are x γ n = a n {displaystyle x_{gamma _{n}}=a_{n}} . Solving the sparsity problem exactly is NP-hard, which is why approximation methods like MP are used. For comparison, consider the Fourier transform representation of a signal - this can be described using the terms given above, where the dictionary is built from sinusoidal basis functions (the smallest possible complete dictionary). The main disadvantage of Fourier analysis in signal processing is that it extracts only the global features of the signals and does not adapt to the analysed signals f {displaystyle f} . By taking an extremely redundant dictionary, we can look in it for atoms (functions) that best match a signal f {displaystyle f} . If D {displaystyle D} contains a large number of vectors, searching for the most sparse representation of f {displaystyle f} is computationally unacceptable for practical applications.In 1993, Mallat and Zhang proposed a greedy solution that they named 'Matching Pursuit.'For any signal f {displaystyle f} and any dictionary D {displaystyle D} , the algorithm iteratively generates a sorted list of atom indices and weighting scalars, which form the sub-optimal solution to the problem of sparse signal representation. In signal processing, the concept of matching pursuit is related to statistical projection pursuit, in which 'interesting' projections are found; ones that deviate more from a normal distribution are considered to be more interesting. Matching pursuit has been applied to signal, image and video coding, shape representation and recognition, 3D objects coding, and in interdisciplinary applications like structural health monitoring. It has been shown that it performs better than DCT based coding for low bit rates in both efficiency of coding and quality of image.The main problem with matching pursuit is the computational complexity of the encoder. In the basic version of an algorithm, the large dictionary needs to be searched at each iteration. Improvements include the use of approximate dictionary representations and suboptimal ways of choosing the best match at each iteration (atom extraction).The matching pursuit algorithm is used in MP/SOFT, a method of simulating quantum dynamics. MP is also used in dictionary learning. In this algorithm, atoms are learned from a database (in general, natural scenes such as usual images) and not chosen from generic dictionaries.