<title>Multiresolution analysis of digital images using the continuous extension of discrete group transforms</title>

Abstract

A new technique is presented for multiresolution analysis (MRA) of digital images. In 2D, it has four variants, two of which are applicable on square lattices; the Discrete Cosine Transform (DCT) is the simpler of the two. The remaining variants can be used in the same way on triangular lattices. The property of the Continuous Extension of the Discrete Group Transform (CEDGT) is used to analyse data for each level of decomposition. The MRA principle is obtained by increasing the data grid for each level of decomposition, and by using an adapted low filter to reduce some irregularities due to noise effect. Compared to some stationary wavelet transforms, the image analysis with a multiresolution CEDGT transform gives better results. In particular, a wavelet transform is capable of providing a local representation at multiple scales, but some local details disappear due to the use of the low pass filter and the reduction of the spatial resolution for a high level of decomposition. This problem is avoided with CEDGT. The smooth interpolation, used by the multiresolution CEDGT, gives interesting results for coarse-to-fine segmentation algorithm and others analysis processes.

Alternating group and multivariate exponential functions

“…, λ n ) be such that λ 1 > λ 2 > · · · > λ n . Then due to relations (17) and (18) in [1] and formulas (13) and (14) we derive that…”

Antisymmetric Orbit Functions

“…In the same way as the Fourier transform leads to discrete Fourier transform, the symmetric orbit function transform leads to a discrete analogue of this transform (which is a generalization of the discrete cosine transform [12]). This discrete transform is useful in many things related to discretization (see [13,14,15,16,17,18,19,20]). Construction of the discrete orbit function transform is fulfilled by means of the results of paper [21].…”

“…This discrete transform is useful in many things related to discretization (see [13,14,15,16,17,18,19,20]). Construction of the discrete orbit function transform is fulfilled by means of the results of paper [21].…”

Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

“…Its various mathematical applications are found in [4,5,6,7,8,9,10,11,12,13,14,15,16]. For other applications, see [15,16,17,18,19,20]. Existence of the continuous version of the transform is implied there as well.…”

“…It is discrete transforms that primarily interest us here because they have a number of practically useful properties [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In particular, continuous extension of the discrete transforms smoothly interpolate digital data in any dimension and for any lattice symmetry afforded by the structure of the given Lie group G. Many examples show that relative to the amount of available data, these transforms provide much smoother interpolation than the conventional Fourier transform.…”