# Rank reduction via MSSA

Consider a block of 3D data of by by samples . The MSSA (Oropeza and Sacchi, 2011) operates on the data in the following way: first, MSSA transforms into of complex values of the frequency domain. Each frequency slice of the data, at a given frequency , can be represented by the following matrix:

 (1)

To avoid notational clutter we omit the argument . Second, MSSA constructs a Hankel matrix for each row of ; the Hankel matrix for row of is as follows:

 (2)

Then MSSA constructs a block Hankel matrix for :

 (3)

The size of is , , . and are predifined integers chosen such that the block Hankel matrices and are close to square matrices. The transformation of the data matrix into a block Hankel matrix can be represented in operator notation as follows:

 (4)

where denotes the Hankelization operator.

In general, the block Hankel matrix can be represented as:

 (5)

where and denote the block Hankel matrix of signal and of random noise, respectively.

We assume that and have full rank, = and has deficient rank, . The singular value decomposition (SVD) of can be represented as:

 (6)

where () and ( ) are diagonal matrices and contain, respectively, larger singular values and smaller singular values. (), ( ), () and ( ) denote the associated matrices with singular vectors. The symbol denotes the conjugate transpose of a matrix. In general the signal is more energy-concentrated and correlative than the random noise. Thus, the larger singular values and their associated singular vectors represent the signal, while the smaller values and their associated singular vectors represent the random noise. We let be to achieve the goal of attenuating random noise while recovering the missing data during the first iteration in reconstruction process as follows:

 (7)

Equation 7 is referred to as the TSVD, which is used in the conventional MSSA approach.

2020-03-10