On Computational Aspects of Krawtchouk Polynomials for High Orders

Abstract

Discrete Krawtchouk polynomials are widely utilized in different fields for their remarkable characteristics, specifically, the localization property. Discrete orthogonal moments are utilized as a feature descriptor for images and video frames in computer vision applications. In this paper, we present a new method for computing discrete Krawtchouk polynomial coefficients swiftly and efficiently. The presented method proposes a new initial value that does not tend to be zero as the polynomial size increases. In addition, a combination of the existing recurrence relations is presented which are in the n- and x-directions. The utilized recurrence relations are developed to reduce the computational cost. The proposed method computes approximately 12.5% of the polynomial coefficients, and then symmetry relations are employed to compute the rest of the polynomial coefficients. The proposed method is evaluated against existing methods in terms of computational cost and maximum size can be generated. In addition, a reconstruction error analysis for image is performed using the proposed method for large signal sizes. The evaluation shows that the proposed method outperforms other existing methods.

Vision-Based Human Detection Techniques: A Descriptive Review

“…Krawtchouk polynomials are utilized for the formation of discrete moments that can illustrate a picture locally [23]. Discrete Krawtchouk polynomials are commonly used in various areas for their exceptional localization property and characteristics [130].…”

“…and fulfills the orthogonality condition Krawtchouk polynomials consistently perform better compared to other polynomials for reconstruction error [23], [130].…”

“…A particular window function is required for signal. The computational time of krawtchouk polynomials is relatively high [23].…”

“…If k = 2 (shown in Figure 9c), (even number) it is difficult to classify the object because of achieving the same score of two classes labels. Orthogonal moments face recognition [143], object classification [144], [145], object recognition [128], [146], texture retrieval [147] • strong signal descriptors with low order elements [129] • computationally expensive [129] Krawtchouk polynomials object recognition [130], edge detection [148], object classification [149], image recognition [150] • better performance for reconstruction error [23], [130] • high computational time [23] Tchebichef polynomials image analysis [131], face Recognition [151], edge detection [132], image retrieval [152] • eliminates the necessity for numerical approximation in the image discrete domain [131], [133] • vulnerable coefficients' calculation to numerical instability for higher polynomial order [132] Charlier polynomials object recognition [153], image classification [154], image reconstruction [155], object recognition [156] • minimizes both the time of computation and the error of propagation [136] • coefficient's numerical inconsistency for higher-order polynomials [136] SKTP face detection [140] • stable in noisy environments [140] • no clear information for handling major occlusion [140] B. DECISION TREE Decision trees are developed for classification tasks.…”

“…The next task is to extract the features and add it to the classification model. Features [23] can be represented by any object properties such as colours, edges, corners, blobs or regions and ridges. The success of the training process depends on the feature extraction, classifier selection and training step (i.e.…”