Analysis and interpretation of an image which was acquired by a nonideal imaging system is the key problem in many application areas. The observed image is usually corrupted by blurring, spatial degradations, and random noise. Classical methods like blind deconvolution try to estimate the blur parameters and to restore the image. In this paper, we propose an alternative approach. We derive the features for image representation which are invariant with respect to blur regardless of the degradation PSF provided that it is centrally symmetric. As we prove in the paper, there exist two classes of such features: the first one in the spatial domain and the second one in the frequency domain. We also derive so-called combined invariants, which are invariant to composite geometric and blur degradations. Knowing these features, we can recognize objects in the degraded scene without any restoration. Index Terms---Degraded image, symmetric blur, blur invariants, image moments, combined invariant...

In this paper, we describe a variational framework for the tomographic reconstruction of an image from the maximum likelihood (ML) estimates of its orthogonal moments. We show how these estimated moments and their (correlated) error statistics can be computed directly, and in a linear fashion from given noisy and possibly sparse projection data. Moreover, thanks to the consistency properties of the Radon transform, this two-step approach (moment estimation followed by image reconstruction) can be viewed as a statistically optimal procedure. Furthermore, by focusing on the important role played by the mmnents of projection data, we immediately see the close connection between tomographic reconstruction of nonnegativevalued images and the problem of nonparametric estimation of probability densities given estimates of their moments. Taking advantage of this connection, our proposed variational algorithm is based on the minimization of a cost functional composed of a term measuring the divergence between a given prior estimate of the image and the current estimate of the image and a second quadratic term based on the error incurred in the estimation of the moments of the underlying image from the noisy projection data. We show that an iterative refinement of this algorithm leads to a practical algorithm for the solution of the highly complex equality constrained divergence minimization problem. We show tbat this iterative refinement results in superior reconstructions of images from very noisy data as compared with the classical filtered back-projection (FBP) algorithm.

The problem of the independence and completeness of rotation moment invariants is addressed in this paper. First, a general method for constructing invariants of arbitrary orders by means of complex moments is described. As a major contribution of the paper, it is shown that for any set of invariants there exists a relatively small basis by means of which all other invariants can be generated. The method how to construct such a basis and how to prove its independence and completeness is presented. Some practical impacts of the new results are mentioned at the end of the paper.

This paper addresses the gray-level image representation ability of the Fourier-Mellin Transform (FMT) for pattern recognition, reconstruction and image database retrieval. The main practical difficulty of the FMT lies in the accuracy and efficiency of its numerical approximation and we propose three estimations of its analytical extension. Comparison of these approximations is performed from discrete and finite-extent sets of Fourier-Mellin harmonics by means of experiments in: (i) image reconstruction via both visual inspection and the computation of a reconstruction error; and (ii) pattern recognition and discrimination by using a complete and convergent set of features invariant under planar similarities. Experimental results

This tutorial aims to present a survey of recent as well as traditional object recognition/classification methods based on image moments. We review various types of moments (geometric moments, complex moments, Legendre moments, Zernike and Pseudo-Zernike moments, and Fourier-Mellin moments) and moment-based invariants with respect to various image degradations and distortions (rotation, scaling, affine transform, image blurring, etc.) which can be used as shape features for classification. We explain a general theory how to construct these invariants and show also a few of them in explicit forms. We review efficient numerical algorithms that can be used for moment computation. Finally, we demonstrate practical examples of using moment invariants in real applications from the area of vision, remote sensing, and medical imaging. The target audience of the tutorial are • researchers from all application areas who need to recognize 2-D objects extracted from binary/graylevel/color images and who look for invariant and robust object features, • specialists in moment-based pattern recognition interested in new development on this field.