Optimal filters for gradient-based motion estimation

Abstract

Extended AbstractEstimating motion between two images plays a vital role in many applications and has drawn a lot of attention during the last two decades. There are many ways to approach this problem and indeed many algorithms have been proposed for this task, e.g. [2, 3, 11. In Barron et. al.[l] a comparative survey of many motion estimation techniques is given. One family of such algorithms which was found to perform well is the family of gradient-based methods, originally proposed by Horn and Schunck [2].The gradient-based methods emerge from the assumption that the intensity value of a physical point in a scene does not change along the image sequence. Denoting the intensity values of the image sequence by the function I(z, y, t ) , where (z, y) is the spatial position and t is the temporal axis, the brightness constancy assumption along the image stream yields [2]:Defining (U", uY) = sequence, we obtain (%, g), as the spatial velocity of each spatio-temporal point in the image I,u" + IYUY + It = 0 .(2) Here I,, Iv and It denote the spatial and temporal derivatives. This Brightness Constancy Equation (BCE), relates the spatial and temporal gradients of an image sequence to the motion vector ( t i z , u y ) at each location (x, y, t ) .One issue that is critical to the implementation of the above BCE is that image derivatives are computed based on sampled information. It is commonly agreed [l, 41 that approximating the spatio-temporal derivatives by finite differences produces error in the above equation and subsequently in the estimated motion. In their comparison study on gradient-based motion estimation, Barron et. [l] conclude that "the method of numerical differentintion is very important -diflerences between first order pixel differencing and higher order central differences are very noticeable" . Several attempts to define or design these derivative filters, together or separately, have been reported in the literature [l, 41. All these methods treat the above question as a problem of optimally designing gradient operators, overlooking the fact that these gradients are to be used for motion estimation. In this paper we first propose a technique to derive a set of filters which are optimal directly with respect to this goal. These filters are designed to give the best estimation results in term of accuracy, where their derivative characteristics are not a demand but a by-product. In our scheme the following requirements are met:'HP

On the Choice of Band-Pass Quadrature Filters

“…Consequently, throughout the history of digital image processing, smoothing and differentiation have been subjects of intense study. A variety of optimal differential operators have been proposed to solve different computer vision problems (see for example: [8,13,14,37,53,63,68,69,78,79] for edge and line detection and [2,16,71] for optical flow estimation). Differentiation is highly sensitive to noise, but can be reduced/avoided using an appropriate scale selection for the smoothing function.…”

Analysing superimposed oriented patterns

“…Nonetheless, future work will, e.g., evaluate the accuracy and noise resistance of other, optimized filters, such as in [43], [44]. Another topic for future research is to improve the segmentation into regions with different number of orientations by adaptive thresholding, for instance based on a Markov random field model [45].…”

“…The values of cos β estimated for each pixel according to (44) are shown in the gray level plots of the middle row. Evidently, there is a slight sensitivity to noise.…”

Bias minimizing filter design for gradient-based image registration

“…These measurements are invariably obtained by application of simple, linear phase, shift invariant filters. Even though these filters play a vital role in the estimation scheme, and have been shown to affect estimator bias [9,2,11], relatively few researchers have studied the design of such filters [12,4,5]. While many papers acknowledge the errors incurred by such gradient approximation schemes, they treat these errors as random in nature and construct statistically robust estimators to minimize their effect.…”