Удосконалення математичної моделі прихованого марковського дерева в задачах попередньої обробки та компресії зображень

Диссертация посвящена усовершенствованию математических моделей и методов предварительной обработки и компрессии, базирующихся на статистическом описании данных в вейвлет-пространстве с использованием аппарата скрытых марковских моделей. Для достижения поставленной цели развита теория описания изображений в вейвлет-пространстве с помощью модели скрытого марковского дерева. Усовершенствованы методы определения параметров модели и методы описания статистических характеристик отдельных вейвлет-коэффициентов, что дало возможность повысить качество работы методов фильтрации изображений. Проведён анализ свойств модели и её разновидностей, что дало возможность улучшить методику сегментации полутоновых и цветных текстурных изображений, а также дало возможность использовать аппарат скрытого марковского дерева в задачах компрессии данных с потерями. Результаты числового моделирования предложенных методов фильтрации, сегментации и компрессии показали эффективность сделанных модификаций.

Ключевые слова: модель скрытого марковского дерева, фильтрация, сегментация, компрессия, вейвлет-преобразование, текстурные изображения.

Kapshiy O.V. Improvement of the hidden markov tree model in images preprocessing and compression tasks. - Manuscript.

Thesis for a candidate's degree in technical sciences by speciality 01.05.02 - Mathematical Modeling and Methods of Calculation. - Lviv Polytechnic National University, Lviv, 2006.

The thesis is dedicated to improvement of mathematical models and methods of preprocessing and lossy compression of images based on theory of hidden markov models. In order to achieve the set goal a theory of image description in wavelet-domain by hidden markov tree model has been evolved. An analysis of the model has been carried out and its defects have been marked. The main shortcomings of the model are: long time of parameters values determination using iterative expectation-maximization algorithm and lack of direct consideration of remain correlation dependencies among neighbour coefficients of same wavelet-transform level.

The first shortcoming has been removed using proposed method of determination of initial approximation of model parameters' values. The method is based on approximation of gaussian distribution mixture, which is a base of hidden markov tree model and describes a probability distribution function of wavelet-coefficients, by a generalized gaussian distribution. According to the method, parameters of gaussians mixture are determined by minimization of relative entropy of the distributions when parameters of generalized gaussian distribution are known, because they can be determined using known values of wavelet-coefficients. Probabilities of model's hidden variables states transitions are determined as an evaluation of frequencies of corresponding states combinations. The variable's states are determined using maximum a posteriori probability minimization. The proposed method allows the long iterative process of model's parameters values determination to be removed at insignificant influence at image modeling quality or its time cost to be considerably decreased. The proposed method of approximation of gaussians mixture by generalized gaussian distribution allows also an entropy of gaussians mixture with known parameters to be estimated. As the estimated value generalized gaussian distribution entropy is used.

In order to take into account remain correlation dependencies among neighbour coefficients at same wavelet-subband when hidden markov tree model is used, the model structure change has been proposed. The change touches the model with known parameters at a stage when the model is used to data processing. At the stage a pseudo-tree based on current coefficient neighbourhood is built. At the pseudo-tree, hidden variables of each neighbour coefficient are connected with hidden variable of current coefficient ancestor at the less detailed wavelet-decomposition level. The pseudo-tree is used when a posteriori probabilities of hidden variables are determined. Changes also touch variances of mixture gaussian distributions. It has been proposed refuse the same values of the variances and use different variances for each wavelet-coefficient. Now, variances are determined using not all wavelet-subband coefficients, but only some neighbourhood. In order to compensate decrease of number of data samples used for variances determination an a priori restrictions in the form of Rayleigh law are used. Using of the proposed model change methods allows method of filtration of images corrupted by additive white gaussian noise to be improved.

Further analysis of image processing methods based on hidden markov tree model has been allowed image filtration method based on a local contextual hidden markov tree model to be improved. Improvement is based on including information about geometrical structure of images into contextual variables of the model. In order to extract information about image geometry a directional Gabor filter is used. The filter is applied to contour images at each wavelet-transform level. In order to avoid renewal of a scaled input image at each wavelet-level by carrying out inverse wavelet-transform and taking into account the fact that information about image geometrical structure is concentrated in high-frequency wavelet-subbands, it has been proposed to built contour images for each level by carrying out partial inverse wavelet-transform. It is assumed, in the proposed method, that values of low-frequency subbands coefficients at processed level are zero. As a result, at each level, a contour image, ready for extracting directional information, is built.

The method of preclassification of image points in a task of grayscale texture images segmentation has been improved. As a classification criterion, the method uses a probability of appearance of wavelet-coefficients for hidden markov tree model with parameters trained for an input image. A new feature vector, which takes into account differences among coefficients from different wavelet-subbands, has been proposed. Possibilities of use the segmentation methods for solving tasks of segmentation of color texture images have been analyzed. The improved method, which takes into account remain dependencies among wavelet-coefficients from different color bands has been proposed.

Proposed method of hidden markov tree model structure change, for considering additional dependencies among neighbourhood coefficients, and proposed method of gaussians mixture entropy estimation have been allowed a method of reverse classification of wavelet-coefficients in a lossy image compression task to be built. The method use allows a result of compression algorithms work to be improved. Ways of increasing of compression algorithm speed have been shown.

Results of numerical modeling of proposed filtration, segmentation and compression methods have shown effectiveness of introduced changes.

Key words: hidden markov tree model, filtration, segmentation, compression, wavelet-transform, texture images.

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