The multiscale Hermite transform for local orientation analysis

Abstract

The efficient representation of local differential structure at various resolutions has been a matter of great interest for adaptive image processing and computer vision tasks. In this paper, we derive a multiscale model to represent natural images based on the scale-space representation: a model that has an inspiration in the human visual system. We first derive the one-dimensional case and then extend the results to two and three dimensions. The operators obtained for analysis and synthesis stages are derivatives of the Gaussian smoothing kernel, so that, for the two-dimensional case, we can represent them either in a rotated coordinate system or in terms of directional derivatives. The method to perform the rotation is efficient because it is implemented by means of the application of the so-called generalized binomial filters. Such a family of discrete sequences fulfills a number of properties that allows estimating the local orientation for several image structures. We also define the discrete counterpart in which the coordinate normalization of the continuous case is approximated as a subsampling of the discrete domain.

A multi-resolution approach for filtering LiDAR altimetry data

“…The signal is analyzed by convolution with a bank of Gaussian derivative filters. 1 Here, we summarize the theory for two-dimensional discrete signals and refer the reader to the original for more details concerning the decomposition of continuous signals (see Silván-Cárdenas and Escalante-Ramírez, 2006, and reference cited therein).…”

“…If zeroes replace residuals up to a given level of the pyramid, the re-synthesized signal is a Gaussiansmoothed version of the original signal. Moreover, any Gaussian-smoothed version of the original signal can be reconstructed by the proper weighting of the coefficients (Silván-Cárdenas and Escalante- Ramírez, 2006). The set of all Gaussian-smoothed signals is termed the scale-space representation of the input signal (Witkin, 1984).…”

A Perceptive Approach to Digital Image Watermarking Using a Brightness Model and the Hermite Transform

“…Moreover, the HT incorporates the Gaussian derivative model of early vision [19][20][21] that considers the derivatives of Gaussian functions as suitable models for ganglionic and cortical visual cells. Similar to DWT, the HT also decomposes the image in a number of coefficients, where zero order coefficients represent a Gaussian-weighted image average.…”

New Families of Fourier Eigenfunctions for Steerable Filtering

“…An important family of wavelets is given by the Hermite-Gauss filters (HGFs) [2], [3], which are defined as the product of Hermite polynomials with an isotropic Gaussian term. HGFs have nice mathematical properties: they are closely related to the Gaussian derivatives, they are very close to the optimal for feature detection [4], and they have interesting causality properties for regularization in the scale space [5], [6].…”