Krawtchouk Moments Research Articles

Privacy-Preserving Krawtchouk Moment feature extraction over encrypted image data

Resource-constrained users outsource the massive image data to the cloud to reduce storage and computation overhead locally, but security and privacy concerns seriously hinder the applications of outsourced image processing services. Besides, existing image processing solutions in the encrypted domain still bring high computation overhead, and even lead to characteristic loss. To this end, we propose a Privacy-Preserving Krawtchouk Moment (PPKM) feature extraction framework over encrypted image data by utilizing the Paillier cryptosystem. First, a mathematical framework for PPKM implementation and image reconstruction is presented in the encrypted domain. Then, the detailed expanding factor and upper bound analysis shows that plaintext Krawtchouk moment and plaintext image reconstruction can be realized over encrypted image with PPKM. Furthermore, the computation complexity of PPKM can be significantly reduced with the block-based parallel algorithm. Experimental results verify that the PPKM is feasible and acceptable in practice in terms of image reconstruction capability and image recognition accuracy.

Improved image registration scheme based on multiscale nonlinear Harris and Gaussian–Krawtchouk moment

Image registration is an indispensable operation in image processing. To solve the problem of nonlinear distortion which easily causes low registration property, a registration scheme including a nonlinear scale space, a new discrete orthogonal Gaussian–Krawtchouk (GK) descriptor vector, and a dual similarity measurement of distance-cosine is proposed. Nonlinear scale space is constructed to preserve a lot of important structures for increasing the quantity of the salient keypoints. Meanwhile, considering both blur stability and simple computation, multiscale Harris algorithm on the basis of nonlinear scale space is applied to extract the keypoints. Moreover, according to different scales, a descriptor vector based on multiscale GK invariant moments is presented and applied to the following image matching. Finally, a dual similarity measurement of the initial Euclidean distance and the fine space cosine is proposed for the purpose of enhancing the matching accuracy. The effectiveness of the proposed algorithm has been tested on the standard images with different types of variations. The experimental results show that the performance of feature extraction and the correct matching rate of the proposed scheme are superior to those of other classical methods. The proposed scheme has a better robustness and a higher registration accuracy.

Single-pixel compressive imaging based on the transformation of discrete orthogonal Krawtchouk moments.

A single-pixel compressive imaging technique that uses differential modulation based on the transformation of discrete orthogonal Krawtchouk moments is proposed. In this method, two sets of Krawtchouk basis patterns are used to differentially modulate the light source, then the Krawtchouk moments of the target object are acquired from the light intensities measured by a single-pixel detector. The target image is reconstructed by applying an inverse Krawtchouk moment transform represented in the matrix form. The proposed technique is verified by both computational simulations and laboratory experiments. The results show that this technique can retrieve an image from compressive measurements and the real-time reconstruction. The background noise can be removed by the differential measurement to realize the excellent image quality. Moreover, the proposed technique is especially suitable for the single-pixel imaging application that requires the extraction of the characteristics at the region-of-interest.

Fast computation of Charlier moments and its inverses using Clenshaw’s recurrence formula for image analysis

In this paper, we propose a new fast way to compute both the image Charlier moments and its inverses using Clenshaw’s recurrence formula. Firstly, we present recursive polynomials of Charlier with respect to the order n and with respect to the variable x and then we define Clenshaw’s recurrence formula to improve the consuming time of the proposed algorithm. So, to show the robustness of the proposed method, a comparative study with the classical method is carried out. In fact, the results of the simulations carried out on binary and gray-scale images show the effectiveness of the proposed method in terms of the calculation time of Charlier moments and in terms of image reconstruction capacity with respect to Krawtchouk moments.

A Comparative Study on Discrete Shmaliy Moments and Their Texture-Based Applications

In recent years, discrete orthogonal moments have attracted the attention of the scientific community because they are a suitable tool for feature extraction. However, the numerical instability that arises because of the computation of high-order moments is the main drawback that limits their wider application. In this article, we propose an image classification method that avoids numerical errors based on discrete Shmaliy moments, which are a new family of moments derived from Shmaliy polynomials. Shmaliy polynomials have two important characteristics: one-parameter definition that implies a simpler definition than popular polynomial bases such as Krawtchouk, Hahn, and Racah; a linear weight function that eases the computation of the polynomial coefficients. We use IICBU-2008 database to validate our proposal and include Tchebichef and Krawtchouk moments for comparison purposes. The experiments are carried out through 5-fold cross-validation, and the results are computed using random forest, support vector machines, naïve Bayes, and k-nearest neighbors classifiers.

Medical Image Blind Integrity Verification with Krawtchouk Moments

A new blind integrity verification method for medical image is proposed in this paper. It is based on a new kind of image features, known as Krawtchouk moments, which we use to distinguish the original images from the modified ones. Basically, with our scheme, image integrity verification is accomplished by classifying images into the original and modified categories. Experiments conducted on medical images issued from different modalities verified the validity of the proposed method and demonstrated that it can be used to detect and discriminate image modifications of different types with high accuracy. We also compared the performance of our scheme with a state-of-the-art solution suggested for medical images—solution that is based on histogram statistical properties of reorganized block-based Tchebichef moments. Conducted tests proved the better behavior of our image feature set.

A new fault diagnosis approach for analog circuits based on spectrum image and feature weighted kernel Fisher discriminant analysis.

Analog circuits are one of the most commonly used components in industrial equipment and, therefore, circuit failure may lead to significant causalities and even huge financial losses. To address this problem, this work presents a fault diagnosis method based on spectrum images for analog circuits. Unlike traditional analysis methods in a one-dimensional space, this study employs a computing method in the field of image processing to realize automatic feature extraction and fault diagnosis in a two-dimensional space. The proposed approach mainly involves the following steps. First, the sampling signals are converted into spectrum images by utilizing cross-wavelet transform, which can be further processed by the following image-based feature extraction method. Then, Krawtchouk moment is applied to extract both the global and local features of the spectrum images and finally form the feature vector. Feature weighted kernel Fisher discriminant analysis is then introduced for locating faults. Two typical analog circuits, video amplifier circuit and opamp high-pass filter circuit, are chosen to demonstrate the effectiveness of the proposed method. Simulation results show that the proposed approach based on spectrum images achieves a high accuracy, thus providing a highly effective means to fault diagnosis for analog circuits.

Efficient 3D object classification by using direct Krawtchouk moment invariants

In this paper, we present an efficient set of moment invariants, named Direct Krawtchouk Moment Invariants (DKMI), for 3D objects recognition. This new set of invariants can be directly derived from the Krawtchouk moments, based on algebraic properties of Krawtchouk polynomials. The proposed computation approach is effectively compared with the classical method, which rely on the indirect computation of moment invariants by using the corresponding geometric moment invariants. Several experiments are carried out so as to evaluate the performance of the newly introduced invariants. Invariability property and noise robustness are firstly investigated. Secondly, the numerical stability is discussed. Then, the performance of the proposed moment invariants as pattern features for 3D object classification is compared with the existing Geometric, Krawtchouk, Tchebichef and Hahn Moment Invariants. Finally, a comparative analysis of computational time of these moment invariants is illustrated. The obtained results demonstrate the efficiency and the superiority of the proposed method.

Comprehensive review of continuous and discrete orthogonal moments in biometrics

ABSTRACTOrthogonal moments (OM) are extensively used for image enhancement, representation and comparison. They are basically derived through mutually perpendicular polynomials which may be continuous and discrete. Primary continuous OM include Zernike, Pseudo-Zernike, Legendre and Bessel–Fourier moments and the prominent discrete OM are Krawtchouk and Tchebichef moments. These moments involve minimum information redundancy and therefore can be applied to various fields like face recognition, edge detection, palm print verification and content-based retrieval. This paper presents the comparative review of the moments in the fields of face, palm print and iris recognition. OM yield better results than non-orthogonal moments because they are robust to noise as well as rotation, scaling and transformation invariant. This paper summarizes the work done by various authors in biometrics for both continuous and discrete moments.

Radial Meixner Moment Invariants for 2D and 3D Image Recognition

In this paper, we propose a new set of 2D and 3D rotation invariants based on orthogonal radial Meixner moments. We also present a theoretical mathematics to derive them. Hence, this paper introduces in the first case a new 2D radial Meixner moments based on polar representation of an object by a one-dimensional orthogonal discrete Meixner polynomials and a circular function. In the second case, we present a new 3D radial Meixner moments using a spherical representation of volumetric image by a one-dimensional orthogonal discrete Meixner polynomials and a spherical function. Further 2D and 3D rotational invariants are derived from the proposed 2D and 3D radial Meixner moments respectively. In order to prove the proposed approach, three issues are resolved mainly image reconstruction, rotational invariance and pattern recognition. The result of experiments prove that the Meixner moments have done better than the Krawtchouk moments with and without nose. Simultaneously, the reconstructed volumetric image converges quickly to the original image using 2D and 3D radial Meixner moments and the test images are clearly recognized from a set of images that are available in a PSB database.

Fusing iris and periocular recognition using discrete orthogonal moment-based invariant feature-set

Iris recognition in uncontrolled environment poses a challenge due to occlusion noise, specular reflections and poor resolution. Therefore, periocular recognition has become a popular biometric modality which when used with iris recognition makes the system suitable for high security applications. The paper introduces discrete orthogonal moment-based invariant features: Tchebichef, Krawtchouk and Dual-Hahn moments which provide discriminative features with compact information and minimum redundancy for non-ideal conditions. The proposed techniques are applied on two publicly available iris databases: IITD v1 and UBIRIS v2 and our own PEC, Chandigarh periocular database. Results demonstrate that the moment-based feature-set outperforms existing approaches available in the literature.

Translation and scale invariants of Krawtchouk moments

Krawtchouk moments are discrete orthogonal moments which are effective for image representation. The translation and scale invariants of Krawtchouk moments are achieved either by normalizing the image or by using a combination of the corresponding invariants of geometric moments. However, the derivation of these functions is not based on Krawtchouk polynomials. In this paper, we propose a new method to derive the translation and scale invariants of Krawtchouk moments directly from the Krawtchouk polynomials. The performance of the proposed method is verified using binary characters. Experimental results show that the values of the Krawtchouk moments are invariant under image translation and scale. Furthermore, the speed of the proposed method is faster than conventional methods.

Three dimensional radial Tchebichef moment invariants for volumetric image recognition

The property of rotation, scaling and translation invariant has a great important in 3D image classification and recognition. Tchebichef moments as a classical orthogonal moment have been widely used in image analysis and recognition. Since Tchebichef moments are represented in Cartesian coordinate, the rotation invariance can’t easy to realize. In this paper, we propose a new set of 3D rotation scaling and translation invariance of radial Tchebichef moments. We also present a theoretical mathematics to derive them. Hence, this paper we present a new 3D radial Tchebichef moments using a spherical representation of volumetric image by a one-dimensional orthogonal discrete Tchebichef polynomials and a spherical function. They have better image reconstruction performance, lower information redundancy and higher noise robustness than the existing radial orthogonal moments. At last, a mathematical framework for obtaining the rotation, scaling and translation invariants of these two types of Tchebichef moments is provided. Theoretical and experimental results show the superiority of the proposed methods in terms of image reconstruction capability and invariant recognition accuracy under both noisy and noise-free conditions. The result of experiments prove that the Tchebichef moments have done better than the Krawtchouk moments with and without noise. Simultaneously, the reconstructed 3D image converges quickly to the original image using 3D radial Tchebichef moments and the test images are clearly recognized from a set of images that are available in a PSB database.

Algebraic technique for computationally efficient Hahn moment invariants

In image processing applications such as scene analysis problems, moments are used as image descriptors of three dimensional objects. These moments may be sensitive to several transformations such as translation and scaling. Three-dimensional moment invariants are possible solution to such problems. Recently, Tchebichef and Krawtchouk moments and their invariants are proposed. These moment invariants are obtained using indirect method or image normalisation method. Perfect invariance cannot be achieved when moment invariants are not obtained from their corresponding polynomials. In this paper, the translation and scale invariance of Hahn moments for two and three dimensional symmetrical and non symmetrical images are derived directly from discrete orthogonal Hahn polynomials using algebraical method. This also enhances the computational efficiency in terms of processing time as compared to the method based on geometric moment invariants. The performance of proposed descriptor is evaluated using binary characters and 3D images. Hahn moment which is generalization of Tchebichef and Krawtchouk moment, give better results with translation and scale invariance and classification problems under clean and noisy image conditions.

Radial Hahn Moment Invariants for 2D and 3D Image Recognition

Recently, orthogonal moments have become efficient tools for two-dimensional and three-dimensional (2D and 3D) image not only in pattern recognition, image vision, but also in image processing and applications engineering. Yet, there is still a major difficulty in 3D rotation invariants. In this paper, we propose new sets of invariants for 2D and 3D rotation, scaling and translation based on orthogonal radial Hahn moments. We also present theoretical mathematics to derive them. Thus, this paper introduces in the first case new 2D radial Hahn moments based on polar representation of an object by one-dimensional orthogonal discrete Hahn polynomials, and a circular function. In the second case, we present new 3D radial Hahn moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Hahn polynomials and a spherical function. Further 2D and 3D invariants are derived from the proposed 2D and 3D radial Hahn moments respectively, which appear as the third case. In order to test the proposed approach, we have resolved three issues: the image reconstruction, the invariance of rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Hahn moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and Princeton shape benchmark (PSB) database for 3D image.

Numerically efficient algorithms for anisotropic scale and translation Tchebichef moment invariants

Anisotropic scale and translation invariants (ASTI) for Tchebichef moments have been proposed by Zhu et al. [27]. Since these invariants are derived via the decomposition of Tchebichef polynomials, it is unavoidable that the invariant algorithms inherit the complexities from the Tchebichef polynomials defined in terms of hypergeometric functions. Furthermore, in order to achieve anisotropic scale and translation invariance, the computation of translation invariants and scale invariants need to be performed sequentially. These have turned out to be the bottleneck for the invariant algorithms. Experimental results show that some of the computed ASTI features from symmetric patterns are less accurate. Thus, we would like to extend the work of Zhu et al.[27] to simplify the complexity of the algorithms and further improve the accuracy of the computed features. The three terms recurrence relation of the Tchebichef polynomials has been used to simplify and improve the computational efficiency of the invariant algorithms. Skew transformations are deployed to enhance the numerical accuracy of ASTI for Tchebhcief moments. Our studies show that the skewed features are less sensitive to noise and significantly enhance the accuracy of pattern recognition systems. This has been verified by the experiments on recognition of printed English letters and leaf patterns corrupted by noise and scaled and translated deformations. The simplification of the algorithms using the orthogonal property of basis functions also can be used to simplify more complex invariants like affine invariants of discrete Tchebichief moments. It can also be extended to derive invariants for other orthogonal based moments like Legendre moments, Krawtchouk moments, Hahn moments, etc.

Identification of ISL Alphabets Using Discrete Orthogonal Moments

In this paper, discrete orthogonal moment-based shape features up to 5th order are proposed for Indian sign language (ISL) recognition system. The shape recognition capability of discrete orthogonal moment-based local features is verified on two databases. These include the standard Jochen-Triesch’s database and 26 ISL alphabets. The ISL alphabets are collected on both uniform and complex backgrounds, with variations in position, scale and rotation. The feature-set is increased for 26 ISL alphabets by varying Region of Interest (ROI) and extracting features from each ROI. A minimum possible feature-set with least redundancy is selected that gives the best recognition accuracy. The effect of order and feature dimensionality for different classifiers is studied. Results show that both Dual-Hahn and Krawtchouk moments are found to exhibit user, scale, rotation and translation invariance. Moreover, they have shape identification capability, thus achieving good recognition accuracy.

Radial invariant of 2D and 3D Racah moments

In this paper, we introduce new sets of 2D and 3D rotation, scaling and translation invariants based on orthogonal radial Racah moments. We also provide theoretical mathematics to derive them. Thus, this work proposes in the first case a new 2D radial Racah moments based on polar representation of an object by one-dimensional orthogonal discrete Racah polynomials on non-uniform lattice, and a circular function. In the second case, we present new 3D radial Racah moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Racah polynomials and a spherical function. Further 2D and 3D invariants are extracted from the proposed 2D and 3D radial Racah moments respectively will appear in the third case. To validate the proposed approach, we have resolved three problems. The 2D/ 3D image reconstruction, the invariance of 2D/3D rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Racah moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges rapidly to the original image using 2D and 3D radial Racah moments, and the test 2D/3D images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and PSB database for 3D image.

Robust Reconstruction and Generalized Dual Hahn Moments Invariants Extraction for 3D Images

In this paper, we introduce a new set of 3D weighed dual Hahn moments which are orthogonal on a non-uniform lattice and their polynomials are numerically stable to scale, consequent, producing a set of weighted orthonormal polynomials. The dual Hahn is the general case of Tchebichef and Krawtchouk, and the orthogonality of dual Hahn moments eliminates the numerical approximations. The computational aspects and symmetry property of 3D weighed dual Hahn moments are discussed in details. To solve their inability to invariability of large 3D images, which cause to overflow issues, a generalized version of these moments noted 3D generalized weighed dual Hahn moment invariants are presented where whose as linear combination of regular geometric moments. For 3D pattern recognition, a generalized expression of 3D weighted dual Hahn moment invariants, under translation, scaling and rotation transformations, have been proposed where a new set of 3D-GWDHMIs have been provided. In experimental studies, the local and global capability of free and noisy 3D image reconstruction of the 3D-WDHMs has been compared with other orthogonal moments such as 3D Tchebichef and 3D Krawtchouk moments using Princeton Shape Benchmark database. On pattern recognition using the 3D-GWDHMIs like 3D object descriptors, the experimental results confirm that the proposed algorithm is more robust than other orthogonal moments for pattern classification of 3D images with and without noise.

Automatic Target Recognition of Military Vehicles With Krawtchouk Moments

The challenge of automatic target recognition of military targets within a synthetic aperture radar scene is addressed in this paper. The proposed approach exploits the discrete-defined Krawtchouk moments, which are able to represent a detected extended target with few features, allowing its characterization. The proposed algorithm provides robust performance for target recognition, identification, and characterization, with high reliability in the presence of noise and reduced sensitivity to discretization errors. The effectiveness of the proposed approach is demonstrated using the MSTAR dataset.