Differentiation by integration using orthogonal polynomials, a survey

This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing.Comment: 35 pages, 3 figures; minor corrections, subsection 3.11 added; accepted by J. Approx. Theor

Fractional order differentiation by integration with Jacobi polynomials

The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises

Multiresolution Moment Filters: Theory and Applications

We introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales. When the window function satisfies a two-scale relation, we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet-like algorithm. We show that B-splines are well-suited window functions because, in addition to being refinable, they are positive, symmetric, separable, and very nearly isotropic (Gaussian shape). We present three applications of these multiscale local moments. The first is a feature-extraction method for detecting and characterizing elongated structures in images. The second is a noise-reduction method which can be viewed as a multiscale extension of Savitzky-Golay filtering. The third is a multiscale optical-flow algorithm that uses a local affine model for the motion field, extending the Lucas-Kanade optical-flow method. The results obtained in all cases are promising

Three-dimensional Acceleration Measurement Using Videogrammetry Tracking Data

In order to evaluate the feasibility of multi-point, non-contact, acceleration measurement, a high-speed, precision videogrammetry system has been assembled from commercially-available components and software. Consisting of three synchronized 640 X 480 pixel monochrome progressive scan CCD cameras each operated at 200 frames per second, this system has the capability to provide surface-wide position-versus-time data that are filtered and twice-differentiated to yield the desired acceleration tracking at multiple points on a moving body. The oscillating motion of targets mounted on the shaft of a modal shaker were tracked, and the accelerations calculated using the videogrammetry data were compared directly to conventional accelerometer measurements taken concurrently. Although differentiation is an inherently noisy operation, the results indicate that simple mathematical filters based on the well-known Savitzky and Golay algorithms, implemented using spreadsheet software, remove a significant component of the noise, resulting in videogrammetry-based acceleration measurements that are comparable to those obtained using the accelerometers

SUBMITTED TO IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Multiresolution Moment Filters: Theory and Applications

Abstract We introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales. When the window function satisfies a two-scale relation, we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution waveletlike algorithm. We show that B-splines are well suited window functions because, in addition to being refinable, they are positive, symmetric, separable, and very nearly isotropic (Gaussian shape). We present three applications of these multi-scale local moments. The first is a feature extraction method for detecting and characterizing elongated structures in images. The second is a noise reduction method which can be viewed as a multi-scale extension of Savitzky-Golay filtering. The third is a multi-scale optical flow algorithm that uses a local affine model for the motion field, extending the Lucas-Kanade optical flow method. The results obtained in all cases are promising