Relationship between the Haar transform and the MRT

Abstract

The Haar transform is an important signal transform that converts real input to real output, and it has been applied in various tasks in signal and image processing. The M-dimensional Real Transform (MRT) is a recently developed transform, and it shares the real-to-real conversion property of the Haar transform. This paper attempts to study the relationships between these two transforms.

A New Integer-to-Integer Transform

“…This structure of the UMRT definition has a similar form as that of the Haar transforms [10]. Table 3 shows the mapping for N ¼ 8.…”

“…From Eqs. (10) and (11) and the definition of MRT in Eq. (1), ðÞ solutions for n in the range 0, N À 1 ½ , and thus, there exist gk , N ðÞ elements in the positive set.…”

“…The Haar transform is related to the Walsh-Hadamard transform [9]. The MRT is related [10] to the Haar transform. One can be obtained from the other through a sequence of bit reversal operations on the rows and columns.…”

“…One can be obtained from the other through a sequence of bit reversal operations on the rows and columns. Similarly, the Hadamard transform and the MRT can be derived from each other [11]. In [12], 2-D gcd-delta functions that contain only zeroes and ones are used to generate integer 2D DFT pairs.…”

Fast Jacket-Haar Transform with Any Size

“…Haar matrix, which is useful for localized signal analysis [19], edge detection [20], OFDM, and filter design and electrocardiogram (ECG) analysis, has been generalized for Jacket-Haar matrix [21], whose entries are 0 and ±2 compared with entries of the original Haar matrix being 1, −1, and 0. Although the 2 -point Jacket-Haar matrices are successfully proposed in [21], there is still a problem on how to construct the arbitrary-length Jacket-Haar transform as the arbitrary-length Walsh-Jacket transform has already done with high efficiency [22].…”