Orthogonal Moments Research Articles

PLCOM: Privacy-preserving outsourcing computation of Legendre circularly orthogonal moment over encrypted image data

In the image outsourcing system, image privacy is still an increasing concern since the cloud service provider and image owners are not in the same trusted domain. The most straightforward method for guaranteeing image privacy is to leverage cryptographic tools, but traditional cryptographic tools make feature extraction algorithms useless. To this end, we propose a privacy-preserving feature extraction scheme for Legendre circularly orthogonal moment, which is a novel global feature descriptor and can be used for image analysis. We first develop a novel feature descriptor, which is one of the circularly orthogonal moments and termed as Legendre Circularly Orthogonal Moment (LCOM). Then, we present a mathematical framework for implementing Privacy-preserving Legendre Circularly Orthogonal Moment (PLCOM) by combining LCOM and somewhat homomorphic encryption, and implement the image reconstruction in the encrypted domain based on PLCOM. Besides, the detailed theoretical analysis of message space and expanding factor generated by the quantitative technology shows that LCOM and image reconstruction in the plaintext domain can be realized with the aid of PLCOM. Finally, experimental results verify that the PLCOM’s performance in terms of image reconstruction capability and image recognition accuracy is acceptable.

Image analysis using modified exponent-Fourier moments

Classic exponent-Fourier moments (EFMs) have been popularly used for image reconstruction and invariant classification. However, EFMs lack natively the translation and scaling-invariant; in addition, they exhibit two types of drawbacks, namely numerical instability and reconstruction error, which in turn influence their reconstruction capability and image classification accuracy. This study considers the challenge of defining modified EFMs (MEFMs), which are based on modified exponent polynomials. In our methods, the basis function of traditional EFMs is appropriately modified, and these modified basis functions are used to replace the original ones. The basis function of the proposed moments is composed of piecewise modified exponent polynomials modulated by a variable parameter exponential envelope. Various types of optimal-order moments can be established by slightly adjusting the bandwidth of the modified basis functions. Finally, we extend the rotation-invariant feature of previous works and propose a new method of scaling and rotation-invariant image recognition using the proposed moments in a log-polar coordinate domain. The translation invariance can then be achieved by an image projection operation, which is substituted for the traditional approach based on the calculation of image geometric moments. The experimental results demonstrate that the MEFMs perform better than traditional EFMs and other classic orthogonal moments including the latest image moments in terms of the image reconstruction capability and the invariant recognition accuracy of smoothing filters, in both noise-free and noisy conditions.

Fast computation of inverse Meixner moments transform using Clenshaw’s formula

Orthogonal moments are recognized as useful tools for object representation and image analysis. It was shown that they have better image representation capability than the continuous orthogonal moments. One problem concerning the use of moments is the high computational cost, which may limit where the online computation is required. In this paper, we propose a recursive method based on Clenshaw’s recurrence formula that can be implemented to transform kernels of Meixner moments and its inverse for fast computation. There is no need for the proposed method to compute the Meixner polynomial values of various orders on various data points, where the computational complexity is reduced. Experimental results show that the proposed method performs better than the existing methods in term of computation speed and the effectiveness of image reconstruction capability in both noise-free and noisy conditions.

Image analysis by two types of Franklin-Fourier moments

In this paper, we first derive two types of transformed Franklin polynomial: substituted and weighted radial Franklin polynomials. Two radial orthogonal moments are proposed based on these two types of polynomials, namely substituted Franklin-Fourier moments and weighted Franklin-Fourier moments &#x0028 SFFMs and WFFMs &#x0029, which are orthogonal in polar coordinates. The radial kernel functions of SFFMs and WFFMs are transformed Franklin functions and Franklin functions are composed of a class of complete orthogonal splines function system of degree one. Therefore, it provides the possibility of avoiding calculating high order polynomials, and thus the accurate values of SFFMs and WFFMs can be obtained directly with little computational cost. Theoretical and experimental results show that Franklin functions are not well suited for constructing higher-order moments of SFFMs and WFFMs, but compared with traditional orthogonal moments &#x0028 e.g., BFMs, OFMs and ZMs &#x0029 in polar coordinates, the proposed two types of Franklin-Fourier Moments have better performance respectively in lower-order moments.

Quaternion Harmonic moments and extreme learning machine for color object recognition

The quaternary orthogonal moments have been widely used as color image descriptors owe to their remarkable color and shape information encapsulation capability. Their computation, however, depends on finding the optimal value of a unit pure quaternion parameter, which is done empirically and with no warranty of optimality. We propose a 2D color object recognition method that relies on the quaternion-valued parameter-free disc-harmonic moment invariants (QHMs) fed into the quaternion extreme learning machine (QELM). The role of this latter is to maintain the correlation between the four parts, real and imaginary, of the quaternary descriptor coefficients. Several datasets are used for recognition experiments. We draw the conclusion that: (1) our quaternion-valued QHMs invariants outperform other quaternary moments, (2) the quaternion-valued moment invariants give results better than the modulus-based moment invariants and (3) the QELM yields results better than the state-of-the-art classifiers.

Rotation and scale invariant image watermarking based on polar harmonic transforms

Geometric attacks such as rotation and scale can destroy the synchronization of watermarking, designing a geometric-distortion resilient watermarking is currently a challenging work. In this paper, we propose a robust image watermarking technique based on polar harmonic transforms (PHTs) which are defined in a circle domain. The watermarking embedding process includes three principal steps: Firstly, we explicitly exploit invariant properties of PHTs’ moments to compute orthogonal moment features. Next, according to the order and repetition of PHTs’ moments, steady moments those are not sensitive to geometric attacks are picked out. Finally, watermark embedding is performed through quantization modulation on the selected steady moments, and the watermarked image is reconstructed by combining the original and the compensating image acquired from unmodulated and modulated PHTs’ moments, respectively. Experimental results show that the proposed watermarking approach is robust to both geometric distortions (rotation and scaling) and general signal processing attacks (Gaussian filtering, median filtering, average filtering, Gaussian noise, salt & pepper noise, speckle noise, JPEG compression, etc.) and outperforms state-of-the-art watermarking methods.

Fast Reconstruction of 3D Images Using Charlier Discrete Orthogonal Moments

We propose a new algorithm for accelerating the computation time of Charlier discrete orthogonal moments for three-dimensional (3D) images, based on two fundamental notions: The first is a new representation of 3D images called image cuboid representation (ICR) in which the 3D image is decomposed into a set of cuboids of the same gray level instead of voxels, enabling a considerable reduction in both the amount of treated voxels and the computation time of Charlier moments. The second is a matrix calculation of the Charlier moments instead of direct or recursive calculations. The significant reduction in the computation time for Charlier moments, in combination with the ICR method, motivated the development of this new method for 3D image reconstruction. Simulation results confirm the effectiveness of the proposed method in terms of the calculation time of 3D Charlier moments as well as the speed and quality of image reconstruction.

Tchebichef and Adaptive Steerable Based Total Variation Model for Image Denoising.

Structural information, in particular, the edges present in an image are the most important part that get noticed by human eyes. Therefore, it is important to denoise this information effectively for better visualization. Recently, research work has been carried out to characterize the structural information into plain and edge patches and denoise them separately. However, the information about the geometrical orientation of the edges are not considered leading to sub-optimal denoising results. This has motivated us to introduce in this paper an adaptive steerable total variation regularizer (ASTV) based on geometric moments. The proposed ASTV regularizer is capable of denoising the edges based on their geometrical orientation, thus boosting the denoising performance. Further, earlier works exploited the sparsity of the natural images in DCT and wavelet domains which help in improving the denoising performance. Based on this observation, we introduce the sparsity of an image in orthogonal moment domain, in particular, the Tchebichef moment. Then, we propose a new sparse regularizer, which is a combination of the Tchebichef moment and ASTVbased regularizers. The overall denoising framework is optimized using split Bregman-based multivariable minimization technique. Experimental results demonstrate the competitiveness of the proposed method with the existing ones in terms of both the objective and subjective image qualities.

Robust Optical Flow Estimation Using Tchebichef Moment Invariant Feature

In this paper, orthogonal moment invariant-based features are used to compute 2D optical flow from sequence of images. The gray level of pixel is described in terms of its local neighborhood using Tchebichef moment invariants rather than its individual intensity value. The description is then normalized in order to make it insensitive to intensity fluctuations due to noise or other perturbations such as varying illumination conditions. The principle of conservation of moment invariance is used to derive overdetermined system of 2D motion constraint equations in local neighborhood of each pixel. The velocity field is then estimated using the least square method. Experimental results are performed on sequential video and thermal image frames under varying environmental conditions. The run time, robustness against noisy and varying illumination conditions and rotation invariance of proposed method are compared with already existing moment and non-moment-based techniques.

Fast 3D image reconstruction by cuboids and 3D Charlier’s moments

In this article, we propose a novel approach to accelerate the processing of 3D images by the discrete orthogonal moments of Charlier. The proposed approach is based on two fundamental notions: The first is the acceleration of the computing time of Charlier discrete orthogonal polynomials and moments in the case of the 3D image using digital filters. The second is the description of the 3D image by a set of cuboids of fixed size instead of individual voxels by decomposing the image by cuboids of small sizes to ensure numerical stability. By applying this method, the 3D Charlier moments are calculated from the cuboids instead of the whole image, as the image processing will be locally in each cuboid. This method allows us to speed up the computation time of the moments and to avoid the problem of propagation of digital errors encountered as well when using of digital filters for 3D images of large sizes. The simulation results show the effectiveness of the proposed method in terms of the computation time of the 3D moments of Charlier and in terms of quality of 3D image.

Explicit Separable two dimensional Moment Invariants for object recognition

Image moment invariants has been widely used in the fields of pattern recognition and computer vision, since they are able to represent pattern features independently of geometric transformations. Currently, Separable Moments and their invariants are gaining more interest, due to their capability for combining the basic properties of different orthogonal moments. However, most of the existing separable moment invariants are derived indirectly from the geometric invariants, based on the relationship orthogonal polynomials and the geometric basis. Therefore, in this paper, we propose a direct approach to construct a set of discrete separable Tchebichef-Krawtchouk Moment Invariants which are simultaneously invariant to Rotation, Scaling and Translation transformation, based on the explicit form of the Tchebichef and Krawtchouk polynomials. Consequently, the experimental and theoretical results validate the effectiveness of the proposed method and show their superiority in image classification and pattern recognition in comparison with the existing methods.

Plant disease recognition using fractional-order Zernike moments and SVM classifier

Orthogonal moments are the projections of image functions onto particular kernel functions. They play vital role in digital image feature extraction being rotation, scaling, translation invariant, robust to image noise and contain minimal information redundancy. These moments are derived from statistically independent orthogonal polynomials which can be continuous or discrete. Most of the modern researches have explored integer-order orthogonal moments, but fractional-order moments are in fact superclass of integer order and more efficient but underrated. This paper proposes fractional-order Zernike moments (FZM) along with SVM to recognize grape leaf diseases. Comparative analysis with integer-order Zernike moments along with other feature selection methods has been explored. FZM–SVM-based technique outperforms other state-of-art techniques yielding $$97.34\%$$ at order 30.

Image analysis by efficient Gegenbauer moments computation for 3D objects reconstruction

In this paper, we suggest a new technique for fast computation of Gegenbauer orthogonal moments for the reconstruction of 3D images/objects; a typical comparison of the proposed method with the con...

Influence of Krawtchouk and Charlier moment’s parameters on image reconstruction and classification

The discrete orthogonal moments such as Krawtchouk and Charlier moments have been used successfully in the field of image analysis. These moments are based on discrete orthogonal polynomials that depend on the locality parameters (p for Krawtchouk and a1 for Charlier). However, the choice of the value of these parameters during the image analysis remains a problem without solutions. In the absence of an analytical study to make these choices, we will study experimentally, in this article, the influence of the variation of the parameters of discrete orthogonal polynomials on the analysis and in particular on the reconstruction and classification of the image. The results of the simulation show the influence of the variation of discrete orthogonal moments parameters of Krawtchouk and Charlier on the quality of reconstruction of the binary, grayscale and colors images in order to make the choice of the numerical value of each parameter. This allows us to obtain a range of parameter values that will be exploited in the classification task of image forms using the invariant moments of Krawtchouk and Charlier.

Image analysis by efficient Gegenbauer moments computation for 3D objects reconstruction

In this paper, we suggest a new technique for fast computation of Gegenbauer orthogonal moments for the reconstruction of 3D images/objects; a typical comparison of the proposed method with the conventional ZOA methods shows significant improvements in term of error reduction, image quality and consumption time. Then we compare our new approach with an existing method using Legendre and Zernike moments in the case of 3D image/object. The obtained results prove that Legendre and Zernike moments are slightly better than Gegenbauer, they are still very efficient and gives very good results in terms of MSE and PSNR. But, Zernike moments have higher computational cost than Gegenbauer moments.

2D and 3D Medical Image Analysis by Discrete Orthogonal Moments

The 2D and 3D medical imaging play a very important role in medical diagnosis to detect and isolate various pathologies. The treatment, exploitation and the interpretation of the obtained images are valuable tools to help establish a medical diagnosis. In this context, we will present in this paper an effective method for the analysis of different types of 2D and 3D medical image based on the discrete orthogonal moments of Tchebichef, Krawtchouk, Hahn, Charlier and Meixner. By this method, the 2D and 3D medical images are described by moment descriptors. The use of these moment descriptors has allowed us to significantly reduce the storage space of 2D and 3D medical images and speed up the image transfer process. The results of simulations on different medical images by the five types of moments have shown the ability of these types to improve and speed up the medical image reconstruction.

Orthogonal rotation invariant features for iris and periocular recognition

In a non-ideal scenario, iris recognition becomes challenging due to occlusion noise by eyelashes and eyelids, specular reflections and illumination variations. This limits its applicability to be used in real-time applications. Thus, periocular recognition is used in complementary to iris recognition which refers to the region around eyes including eyelashes, eyelids and skin texture. By fusing both iris and periocular modalities, a more reliable and an accurate biometric system is attained that can be considered for high surveillance applications. The proposed techniques are based on continuous orthogonal moments: Zernike moments and polar harmonic transforms which are invariant to rotation and noise. These capture local intensity variations of the neighbourhood pixels that pertain to shape details of the periocular region and random texture pattern of the iris region. The techniques have been evaluated on iris databases: IITD v1 and UBIRIS v2 and a self-developed PEC, Chandigarh periocular database which has been created in a less constrained environment for the research community working on periocular recognition. Results demonstrate that the proposed technique gives encouraging results in comparison to the existing approaches.

3D Image Representation Using Separable Discrete Orthogonal Moments

This paper introduces a new sets of separable three-dimensional moments, called Racah-Tchebichef-Krawtchouk Moments (RTKM), Racah-Krawtchouk-Krawtchouk Moments (RKKM) and Racah-Racah-Krawtchouk Moments (RRKM), for the field of 3D image representation. First, we will provide the detailed mathematical framework of the proposed separable 3D moments. Second, we outline the main applications of the newly introduced moments. Where, the global feature extraction capability is evaluated by measuring reconstruction quality of 3D image, under noisy and noise-free conditions. While, the ability to extract local feature is also demonstrated by adjusting the polynomials parameters value according to the desired Region of Interest (ROI). The obtained results show that the proposed separable moments perform better than the classical moments in terms of image representation capability.

Image retrieval and classication using shifted Legendre invariant moments and Radial Basis Functions Neural Networks

Since shape is one of the important low-level features of any Image Retrieval and any image classification system, we present in this paper a new set of orthogonal polynomials called shifted Legendre polynomials, this helps to build a set of orthogonal moments, which are invariant to translation, rotation and scale. We apply new image Retrieval and classification systems based on the proposed invariant moments and Radial Basis Functions Neural Networks. To show the effectiveness of our approaches we present some experimental results.

Exact Legendre–Fourier moments in improved polar pixels configuration for image analysis

This study presents the exact Legendre–Fourier moments and a novel arrangement of polar pixels that allows calculating orthogonal moments defined in a unit radius more accurately than traditional methods. This arrangement simplifies implementation and preserves the values of the pixels of the image during the calculation of the moments. Moreover, the exact Legendre–Fourier moments use the weighted substituted radial shifted Legendre polynomials as kernel, which has the ability to accurately calculate the circular moments. Finally, the author presents a comparative analysis of the reconstruction error with existing configurations and other families of circular moments. The results indicate that the mehod provides a significant advantage.