Showing papers on "Kernel (image processing) published in 1974"

Periodic Splines and Spectral Estimation

Abstract: The theory of periodic smoothing splines is presented, with application to the estimation of periodic functions. Several theorems relating the order of the differential operator defining the spline to the saturation (order of bias) of the estimator are proven. The linear operator which maps a function to its periodic continuous smoothing spline approximation is represented as a convolution operator with a given convolution kernel. This operator is shown to be the limit of a sequence of operators which map a function into the periodic version of the usual lattice smoothing spline. The convolution kernel above appears as the kernel in a kernel type estimate of the spectral density. Thus, it is shown that, a smoothing spline spectral density estimate, is also asymptotically a kernel type spectral density estimate. Some numerical results are presented.

Image restoration by discrete convolution of minimal length

Abstract: A conventional method of restoring blur-degraded image data is inverse filtering. This technique is usually implemented with 32 or 64 image data points contributing, via two Fourier transforms or one convolution, to each restored output point. We show here that actually about seven data points (in a seven-term convolution) suffice for producing about the same resolution and side-lobe characteristics as for inverse filtering. We calculate, by use of a computer search routine, the necessary weights for use in a convolution-type restoring formula of 5, 7, 11, or 15 terms. The criterion for weight selection is a required first-zero position in the output point spread function and, simultaneously, a minimum value of the largest side lobe in the point spread function. The 15-term case is further constrained to have weights selected from the values −1, 0, and +1, only. This defines a method of restoring by addition and subtraction of image values. Experimental results using diffraction-blurred edge data are used to test the methods.

On the integral equations of the linear theory of recurrent lateral interaction in vision

Abstract: In a spatially homogeneous visual system with linear, recurrent, lateral interactions that fall off with distance at large distances, the response field is related to the excitation field through a functional equation involving convolution of the response with an “interaction kernel” which characterizes the system. We discuss here the general theory of solutions of such equations in Bochner's F k-spaces, and we present explicit solutions for some special cases in which the interaction kernel is “bimodal,” i.e., achieves its maximum absolute value at a non-zero distance of separation of sensory points. Bimodal kernels are of interest because they appear to be in qualitative accord with physiological data, and, as is known, a linear recurrent system with such a kernel can yield a transfer function whose maximum value is finite and occurs at a finite, non-zero, spatial frequency. Such a recurrent system with a bimodal kernel and a regular transfer function can, however, also show “oscillations,” i.e., multiple turning points, in the response to a simple step-function excitation field. To help study this last phenomenon, we derive resolvent functions for integral equations in a class which generalizes the classical Lalesco-Picard equation.

Improved algorithm for low-order real digital circular convolution

Abstract: A new algorithm, based on the minimisation of add and multiply operations, is presented for the computation of low-order real digital circular convolutions.