Information quantity in a pixel of digital image

The paper is devoted to the problem of integer-valued estimating of information quantity in a pixel of digital image. The definition of an integer estimation of information quantity based on constructing of the certain binary hierarchy of pixel clusters is proposed. The methods for constructing hierarchies of clusters and generating of hierarchical sequences of image approximations that minimally differ from the image by a standard deviation are developed. Experimental results on integer-valued estimation of information quantity are compared with the results obtained by utilizing of the classical formulas.

Hierarchical pixel clustering for image segmentation

In the paper a piecewise constant image approximations of sequential number of pixel clusters or segments are treated. A majorizing of optimal approximation sequence by hierarchical sequence of image approximations is studied. Transition from pixel clustering to image segmentation by reducing of segment numbers in clusters is provided. Algorithms are proved by elementary formulas.

Image segmentation by optimal and hierarchical piecewise constant approximations

Piecewise constant image approximations of sequential number of segments or clusters of disconnected pixels are treated. The method of majorizing of optimal approximation sequence by hierarchical sequence of image approximations is proposed. A generalization for multidimensional case of color and multispectral images is foreseen.

Reclassification formula that provides to surpass K-means method

The paper presents a formula for the reclassification of multidimensional data points (columns of real numbers, "objects", "vectors", etc.). This formula describes the change in the total squared error caused by reclassification of data points from one cluster into another and prompts the way to calculate the sequence of optimal partitions, which are characterized by a minimum value of the total squared error E (weighted sum of within-class variance, within-cluster sum of squares WCSS etc.), i.e. the sum of squared distances from each data point to its cluster center. At that source data points are treated with repetitions allowed, and resulting clusters from different partitions, in general case, overlap each other. The final partitions are characterized by "equilibrium" stability with respect to the reclassification of the data points, where the term "stability" means that any prescribed reclassification of data points does not increase the total squared error E. It is important that conventional K-means method, in general case, provides generation of instable partitions with overstated values of the total squared error E. The proposed method, based on the formula of reclassification, is more efficient than K-means method owing to converting of any partition into stable one, as well as involving into the process of reclassification of certain sets of data points, in contrast to the classification of individual data points according to K-means method.

Stable Segmentation of Digital Image

In the paper the optimal image segmentation by means of piecewise constant approximations is considered. The optimality is defined by a minimum value of the total squared error or by equivalent value of standard deviation of the approximation from the image. The optimal approximations are defined independently on the method of their obtaining and might be generated in different algorithms. We investigate the computation of the optimal approximation on the grounds of stability with respect to a given set of modifications. To obtain the optimal approximation the Mumford-Shuh model is generalized and developed, which in the computational part is combined with the Otsu method in multi-thresholding version. The proposed solution is proved analytically and experimentally on the example of the standard image.