Circular Harmonic Decomposition Research Articles

Rotation Invariant Parallel Signal Processing Using a Diffractive Phase Element for Image Compression

We propose a new rotation invariant correlator using dimensionality reduction. A diffractive phase element is used to focus image data into a line which serves as input for a conventional correlator. The diffractive element sums information over each radius of the scene image and projects the result onto one point of a line located at a certain distance behind the image. The method is flexible, to a large extent, and might include parallel pattern recognition and classification as well as further geometrical invariance. Although the new technique is inspired from circular harmonic decomposition, it does not suffer from energy loss. A theoretical analysis, as well as examples, are given.

Image and feature reconstruction for the attenuated Radon transform via circular harmonic decomposition of the kernel

This paper is concerned with a method of image reconstruction and feature extraction for the attenuated Radon transform in two dimensions based on the decomposition in circular harmonics of the integral kernel in Novikovʼs inversion formula for an arbitrary known attenuation. This analytical decomposition of the reconstruction kernel provides an alternative reconstruction algorithm. Besides, we propose to use our formula to directly extract features of the object with no need for process imaging techniques. Numerical results attest to the strengths and limitations of our reconstruction method in terms of accuracy and robustness for image and feature reconstruction.

Cormack-type inversion of attenuated Radon transform

The reconstruction of a function from its attenuated Radon transform (AtRT) finds many applications in medical and physical imaging. A lot of different algorithms of inversion of AtRT have been described based on the Novikov formula. In this paper, we consider a particular case of the AtRT, when the attenuation is axially symmetric. The reconstruction algorithm is based on circular harmonic decomposition. The work is in fact a natural generalization of the Cormack algorithm for the exponential Radon transform. This method of reconstruction is applicable to axially symmetric attenuated Radon transform with a complex attenuation and can be used for reconstruction in polarized tomography of stress tensor fields.

Novel numerical inversions of two circular-arc radon transforms in Compton scattering tomography

Compton scattering tomography (CST) is an alternative imaging process which reconstructs, in a two-dimensional slice, the electron density of an object by collecting radiation emitted from an external source and scattered throughout this object. The collected data at specific scattering energies appears essentially as the integral of the electron density on definite families of arcs of circles. Reconstruction of the unknown electron density is achieved by the inversion of the corresponding circular-arcs Radon transforms (CART). We review two existing CST modalities, their corresponding CART and establish their numerical inversion algorithms in the formalism of the so-called circular harmonic decomposition (CHD) for a function. The quality of the reconstructed images is illustrated by numerical simulations on test phantoms. Comparison with standard tomography performances demonstrates the efficiency and interest of this inversion method via CHD in imaging science such as biomedical imaging and non-destructive industrial testing.

Radon transforms on generalized Cormack’s curves and a new Compton scatter tomography modality

In his seminal work of 1981, Cormack established that Radon transforms defined on two remarkable families of curves in the plane are invertible and admit explicit inversion formulas via circular harmonic decomposition. A sufficient condition for finding larger classes of curves enjoying the same property is given in this paper. We show that these generalized Cormack’s curves are given by the solutions of a nonlinear first-order differential equation, which is invariant under geometric inversion. A derivation of the analytic inverse formula of the corresponding Radon transforms, as well as some of their main properties, are worked out. Interestingly, among these generalized Cormack’s curves are circles orthogonal to a circle of fixed radius centered at the origin of coordinates. It is suggested that a novel Compton scatter tomography modality may be modeled by a Radon transform defined on these circles.

Multiresolution circular harmonic decomposition

A dictionary of complex waveforms suited for multiresolution analysis and individually steerable by multiplication by a complex factor is presented. It is based on circular harmonic wavelets (CHW) and is useful for pattern analysis under rotations. The main theoretical aspects of CHWs are illustrated, and an example of application to motion estimation is provided.

Rotation-invariant and controllable space-variant optical correlation

We propose a method for designing a correlator for achieving rotation-invariant and controllable space-variant optical correlation. The design concept is based on a combination of fractional correlation and circular-harmonic decomposition of the reference object. The suggested method is described and analyzed in detail. Numerical simulations show that this new correlator might provide potential applications in practice.

Invariant pattern recognition by use of wavelength multiplexing

Rotation-invariant pattern recognition can be achieved with circular-harmonic decomposition. A common problem with such a filter is that, because it is only a single term out of the circular decomposition, it does not contain much of the reference object's energy. Thus, the obtained correlation selectivity is low. This problem is solved by use of wavelength multiplexing. First, different harmonic terms are encoded by different wavelengths, and then they all are added incoherently in the output correlation plane. This process leads to rotation-invariant pattern recognition with a higher discrimination ability.

Improved Matched Filter Based on Radial Eigenfunctions

Abstract A new rotation invariant filter is proposed. The Karhunen-Loeve (KL) expansion is used to find a set of radial eigenfunctions corresponding to a target. A pseudo-object is reconstructed by the inverse KL transform using a reduced number of those functions. The rotation-invariant filter using radial eigenfunctions is obtained with the use of the circular harmonic decomposition of the pseudo-object. A computer simulation shows that this filter gives good results for discrimination and pattern recognition in the presence of a cluttered background.

The circular harmonic Radon transform

The circular harmonic decomposition method for evaluating the inverse Radon transform is investigated. A discrete, finite set of projection data may be aliased and its interpretation is inevitably non-unique. When the inverse Radon transform is approximated by a summation, the filtered back projection, it is shown that as well as being non-unique, the reconstruction is inconsistent with the data. By contrast, the circular harmonic decomposition produces a consistent image. The stable form of the method is used to develop a simple and efficient numerical algorithm. This is illustrated with various simple examples and head phantoms.