Optical median filtering using threshold decomposition

Abstract

A hybrid optical/electronic system performs median filtering and related ranked-order operations using threshold decomposition to encode the image. Threshold decomposition transforms the nonlinear neighborhood ranking operation into a linear space-invariant filtering step followed by a point-to-point threshold comparison step. Spatial multiplexing allows parallel processing of all the threshold components as well as recombination by a second linear space-invariant filtering step. An incoherent optical correlation system performs the linear filtering, using a magnetooptic spatial light modulator as the input device and a computergenerated hologram in the filter plane. Thresholding is done electronically. By adjusting the value of the threshold, the same architecture is used to perform median, minimum, and maximum filtering of images.

Representations for Morphological Image Operators and Analogies with Linear Operators

“…Early examples include cellular automata, parallel architectures, VLSI, optical/electronic, and analog optical implementations of morphological and rank filters whose special cases are the simple erosions and dilations (Preston et al, 1979;Sternberg, 1980;Harber et al, 1985;Ochoa et al, 1987;O'Neil and Rhodes, 1986;Hereford and Rhodes, 1988;Barrera et al, 1994). Nowadays, the vast majority of many computer (hardware or software) systems for digital image processing include among their main operations the basic morphological operators such as erosions, dilations, openings and closings.…”

Sorting continuous-time signals and the analog median filter

“…The need for a device that would perform two-dimensional (2-D) median filtering, included as a stage in the optical section at the front end of the imaging system, is mentioned in [1]. A hybrid optical/electronic system has in fact been described [5], but it is still a step away from the analog, optical median filter. Although the implementation issues are not the main purpose of this paper, we will discuss them briefly in Section VI in the hope of estimulating further research in that important direction.…”

“…This is clear when the sorting of is continuous at , in which case, the absolute value of the difference between the outputs of the median filter and the filter defined by (4) Using similar arguments, it is possible to show that the first integral converges to , whereas the second integral converges to . Thus, when is not continuous, the output of the filter defined by (4) converges to the output of the analog median filter defined by (5) Note that because is left continuous. In fact, the filter described by (5) can also be interpreted as follows: Define the "average sorting" of as the average of its left-continuous and right-continuous sortings, and then redefine the median filter in terms of this "average sorting.…”

“…Thus, when is not continuous, the output of the filter defined by (4) converges to the output of the analog median filter defined by (5) Note that because is left continuous. In fact, the filter described by (5) can also be interpreted as follows: Define the "average sorting" of as the average of its left-continuous and right-continuous sortings, and then redefine the median filter in terms of this "average sorting. "…”

“…Optical implementations, which were asked for in [1], are also a possibility. The hybrid electronic/optical filter described in [5] was a step in this direction. Implementations similar to the one described in [12] may also be of interest: the circuit, which contains no sorting subsystem or clock signal, is based on nonlinear equations, the solution of which leads to the order statistics.…”