Abstract-In this paper, we have derived analytic expressions for the phase correlation of downsampled images. We have shown that for downsampled images the signal power in the phase correlation is not concentrated in a single peak, but rather in several coherent peaks mostly adjacent to each other. These coherent peaks correspond to the polyphase transform of a filtered unit impulse centered at the point of registration. The analytic results provide a closed-form solution to subpixel translation estimation, and are used for detailed error analysis. Excellent results have been obtained for subpixel translation estimation of images of different nature and across different spectral bands.
Recent developments in statistics now allow maximum likelihood estimators for the parameters of Markov random fields (MRFs) to be constructed. We detail the theory required, and present an algorithm that is easily implemented and practical in terms of computation time. We demonstrate this algorithm on three MRF models--the standard Potts model, an inhomogeneous variation of the Potts model, and a long-range interaction model, better adapted to modeling real-world images. We estimate the parameters from a synthetic and a real image, and then resynthesize the models to demonstrate which features of the image have been captured by the model. Segmentations are computed based on the estimated parameters and conclusions drawn.
cation [6,7, 14, 15]. It is well known that multigrid methods can improve significantly the convergence rate and the In this paper, we are interested in massively parallel multiscale relaxation algorithms applied to image classification. quality of the final results of iterative relaxation techniques.
It is well known that multigrid methods can improve signifi-There are many approaches in multigrid image segmencantly the convergence rate and the quality of the final results tation. A well known approach is the renormalization of iterative relaxation techniques. First, we present a classical group algorithm which is based on renormalization group multiscale model which consists of a label pyramid and a whole ideas from statistical physics. This technique has been observation field. The potential functions of coarser grids are adapted by Gidas [13] to image processing. The main adderived by simple computations. The optimization problem is vantage of the method is that it provides a mechanism for first solved at the higher scale by a parallel relaxation algorithm; relating the processing at different scales with one another. potentials, and hence one can use classical relaxation algorithms to minimize the energy at coarser grids. Unfortunately, such approximations are available only for certain
then the next lower scale is initialized by a projection of the
INTRODUCTIONsimple models, mainly in image restoration [13]. Another Markov random fields (MRF) have become more and interesting model has been proposed by Bouman and Shamore popular during the past few years in image processing piro [7]. This model consists of a label pyramid where each [1,5,8,9, 12, 27]. A good reason for this is that MRF level is causally dependent on the coarser layer above it. require less a priori information on the world model. On The model results in a new optimization criterium called the other hand, the local behavior of MRF allows for the sequential MAP estimate. This model yields to a noniteradevelopment of highly parallel algorithms in combinatorial tive segmentation algorithm and direct methods of parameoptimization problems.ter estimation. In this paper, we are interested in massively parallelThe basis of our approach is a consistent multiscale MRF multiscale relaxation algorithms applied to image classifi-model originally proposed by Heitz et al. in [14, 15] for motion analysis. Related models can also be found in [6] for texture segmentation and in [17] for image reconstruction. * This work has been partially funded by CNES (French Space This model consists of a label pyramid and a whole obser-
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