Guo-Niu Han

Blurred Image Recognition by Legendre Moment Invariants

Processing blurred images is a key problem in many image applications. Existing methods to obtain blur invariants which are invariant with respect to centrally symmetric blur are based on geometric moments or complex moments. In this paper, we propose a new method to construct a set of blur invariants using the orthogonal Legendre moments. Some important properties of Legendre moments for the blurred image are presented and proved. The performance of the proposed descriptors is evaluated with various point-spread functions and different image noises. The comparison of the present approach with previous methods in terms of pattern recognition accuracy is also provided. The experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments.

Reconstruction of tomographic images from limited range projections using discrete Radon transform and Tchebichef moments

This paper presents an image reconstruction method for X-ray tomography from limited range projections. It makes use of the discrete Radon transform and a set of discrete orthogonal Tchebichef polynomials to define the projection moments and the image moments. By establishing the relationship between these two sets of moments, we show how to estimate the unknown projections from known projections in order to improve the image reconstruction. Simulation results are provided in order to validate the method and to compare its performance with some existing algorithms.

Image reconstruction from limited range projections using orthogonal moments

A set of orthonormal polynomials is proposed for image reconstruction from projection data. The relationship between the projection moments and image moments is discussed in detail, and some interesting properties are demonstrated. Simulation results are provided to validate the method and to compare its performance with previous works.

General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes

Zernike polynomials have been widely used to describe the aberrations in wavefront sensing of the eye. The Zernike coefficients are often computed under different aperture sizes. For the sake of comparison, the same aperture diameter is required. Since no standard aperture size is available for reporting the results, it is important to develop a technique for converting the Zernike coefficients obtained from one aperture size to another size. By investigating the properties of Zernike polynomials, we propose a general method for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes.

The q-tangent and q-secant numbers via basic Eulerian polynomials

The classical identity that relates Eulerian polynomials to tangent numbers together with the parallel result dealing with secant numbers is given a q-extension, both analytically and combinatorially. The analytic proof is based on a recent result by Shareshian and Wachs and the combinatorial one on the geometry of alternating permutations.

An Efficient Method for Computation of Legendre Moments

The two-dimensional (2D) and three-dimensional (3D) orthogonal moments are useful tools for 2D and 3D object recognition and image analysis. However, the problem of computation of orthogonal moments has not been well solved because there exist few algorithms that can efficiently reduce the computational complexity. As is well known, the calculation of 2D and 3D orthogonal moments by a straightforward method requires a large number of additions and multiplications. In this paper, an efficient algorithm for computing 2D and 3D Legendre moments is presented. First, a new approach is developed for computing Legendre polynomials with one variable; the corresponding results are then used to calculate 1D Legendre moments. Second, we extend our method to calculating 2D Legendre moments, a more accurate approximation formula when an analog original image is digitized to its discrete form is also discussed, and the relationship between the usual approximation and the new approach is investigated. Finally, an efficient method for computing 3D Legendre moments is developed. As one can see, the proposed algorithm improves the computational efficiency significantly and can be implemented easily for high order of moments.

Some Conjectures and Open Problems on Partition Hook Lengths

We present some conjectures and open problems on partition hook lengths motivated by known results on the subject. The conjectures were suggested by extensive experimental calculations using a computer algebra system. The first conjecture unifies two classical results on the number of standard Young tableaux and the number of pairs of standard Young tableaux of the same shape. The second unifies the classical hook formula and the marked hook formula. The third includes the longstanding Lehmer conjecture, which says that the Ramanujan tau function never assumes the value zero. The fourth is a more precise version of the third in the case of 3-cores. We also list some open problems on partition hook lengths.

Signed words and permutations, I: A fundamental transformation

The statistics major index and inversion number, usually defined on ordinary words, have their counterparts in signed words, namely the socalled flag-major index and flag-inversion number. We give the construction of a new transformation on those signed words that maps the former statistic onto the latter one. It is proved that the transformation also preserves two other set-statistics: the inverse ligne of route and the lower records.

Un autre q-analogue des nombres d’Euler

The ordinary generating functions of the secant and tangent numbers have very simple continued fraction expansions. However, the classical q-secant and q-tangent numbers do not give a natural q-analogue of these continued fractions. In this paper, we introduce a different q-analogue of Euler numbers using q-difference operator and show that their generating functions have simple continued fraction expansions. Furthermore, by establishing an explicit bijection between some Motzkin paths and (k,r)-multipermutations we derive combinatorial interpretations for these q-numbers. Finally the allied q-Euler median numbers are also studied.

The Triple, Quintuple and Septuple Product Identities Revisited

This paper takes up again the study of the Jacobi triple and Watson quintuple identities that have been derived combinatorially in several manners in the classical literature. It also contains a proof of the recent Farkas-Kra septuple product identity that makes use only of “smanipulatorics” methods.