Continuous Orthogonal Moments Research Articles

Fast image reconstruction method using radial harmonic Fourier moments and its application in digital watermarking

The Radial Harmonic Fourier Moments(RHFMs) is a kind of continuous orthogonal moments with good performance of image representation and reconstruction. Most of existing methods focused on improving the computation of RHFMs, and ignored the research about the reconstruction. Therefore, a fast reconstruction method based on RHFMs by using inverse fast Fourier transform(IFFT) is proposed in this paper. The time cost of reconstruction is greatly decreased. Then, the fast computation method is extend to the quaternion radial harmonic Fourier moments(QRHFMs) by using quaternion theory, which is suitable for the color image representation. Finally, a color image watermarking scheme based on the QRHFMs is conducted. During the embedding process, considering the association between QRHFMs and quaternion discrete Fourier transform(QDFT), the watermark is embedded in the magnitude of QRHFMs symmetrically. The center area of cover image is ignored in order to improve the quality of watermarked image. Experiments denote that proposed watermarking algorithm has low computation complexity and good robust against geometric attacks and common attacks.

Fast and accurate computation of polar harmonic Fourier moments for image description.

Continuous orthogonal moments are widely used in various image techniques due to their simplicity and good rotational invariance and stability. In recent years, numerous excellent continuous orthogonal moments have been developed, among which polar harmonic Fourier moments (PHFMs) exhibit strong image description capabilities. However, the numerical integration error is large in the calculation, which seriously affects the calculation accuracy, especially in higher-order calculation. In this paper, a continuous orthogonal moments-fast and accurate PHFM (FAPHFM) is proposed. It utilizes the polar pixel tiling technique to reduce numerical errors in the computation; this method particularly improves the accuracy of higher-order moments of traditional PHFMs. However, as accuracy increases, calculation complexity also increases. To address this issue, an eight-way symmetric/anti-symmetric calculation of the angular and radial functions was performed using the symmetry and anti-symmetry of traditional PHFMs, and clustering of pixels was performed as a way to improve the computational speed. The experimental results show that FAPHFMs perform better in image reconstruction (including noise), with higher computational accuracy, lower time complexity, and better image description ability.

Multi-dimensional hypercomplex continuous orthogonal moments for light-field images

Image moments have always been topical in the domain of image processing, and the proposed hypercomplex moments and multi-channel moments have greatly broadened the application of image moments in recent years. However, the hypercomplex moments are limited by the specific number of imaginary parts of the hypercomplex number, and the redundancy of imaginary parts or the overflow of image channels will occur when the number of imaginary parts does not fit the image structure. The multi-channel moments allow flexible adjustment of the number of channels, but their loose structure will result in the loss of correlation between multiple channels of images. Moreover, the hypercomplex moments and multi-channel moments cannot handle images with multi-dimensional structures such as light-field (LF) images. For the purpose of addressing the above problems, this paper proposes generalized hypercomplex continuous orthogonal moments (HCOMs) and extends them to multi-dimensional hypercomplex continuous orthogonal moments (MHCOMs) in multi-dimensional space. MHCOMs can be used to process images with arbitrary multi-dimensional structures, and when they are used for LF image processing, there is no need to map the LF images to two-dimensional space to change the multi-dimensional structural properties of LF images. By calculating MHCOMs of LF images, a set of multi-dimensional hypercomplex tensors containing robust feature information of LF images can be obtained, and the original LF images can be reconstructed using this information. Furthermore, the MHCOMs-based zero-watermarking scheme is designed for LF images, which has excellent robustness and can achieve lossless copyright protection for LF images.

Local Orthogonal Moments for Local Features.

By introducing parameters with local information, several types of orthogonal moments have recently been developed for the extraction of local features in an image. But with the existing orthogonal moments, local features cannot be well-controlled with these parameters. The reason lies in that zeros distribution of these moments' basis function cannot be well-adjusted by the introduced parameters. To overcome this obstacle, a new framework, transformed orthogonal moment (TOM), is set up. Most existing continuous orthogonal moments such as Zernike moments, fractional-order orthogonal moments (FOOMs), etc. are all special cases of TOM. To control the basis function's zeros distribution, a novel local constructor is designed, and local orthogonal moment (LOM) is proposed. Zeros distribution of LOM's basis function can be adjusted with parameters introduced by the designed local constructor. Consequently, locations, where local features extracted from by LOM, are more accurate than those by FOOMs. In comparison with Krawtchouk moments and Hahn moments etc., the range, where local features are extracted from by LOM, is order insensitive. Experimental results demonstrate that LOM can be utilized to extract local features in an image.

Stereoscopic Image Description With Trinion Fractional-Order Continuous Orthogonal Moments

Some research progress has been made on fractional-order continuous orthogonal moments (FrCOMs) in the past two years. Compared with integer-order continuous orthogonal moments (InCOMs), FrCOMs increase the number of affine invariants and effectively improve numerical stability. However, the existing types of FrCOMs are still very limited, of which all are planar image oriented. No report on stereoscopic images is available yet. To this end, in this paper, FrCOMs corresponding to various types of InCOMs are first deduced, and then, they are combined with trinion theory to construct trinion FrCOMs (TFrCOMs) applicable to stereoscopic images. Furthermore, the reconstruction performance and geometric invariance of TFrCOMs are analyzed theoretically and experimentally. Finally, an application in the stereoscopic image zero-watermarking algorithm is investigated to verify the superior performance of TFrCOMs.

Octonion continuous orthogonal moments and their applications in color stereoscopic image reconstruction and zero-watermarking

Continuous orthogonal moments (COMs) are a type of effective image features widely used in various fields of image processing. However, most of the existing COMs are used for processing flat images and are not suitable for color stereoscopic images. For this reason, this paper first proposes an octonion theory applicable to color stereoscopic images, all color components of color stereoscopic images are coded by using the imaginary part of octonion, and all color components are processed as a whole, and the internal relations among all components are preserved. Then this paper combines the octonion theory with COMs to propose the octonion continuous orthogonal moments (OCOMs). The OCOMs fully reflect and retain the specific correlations between the left- and right-view components of color stereoscopic images, and provide good image description capability. Experimental results show that OCOMs have strong stability and good reconstruction performance when processing color stereoscopic images. Compared with other zero-watermarking methods, the zero-watermarking method embedded by OCOMs has stronger robustness.

Color image analysis with novel trinion decimal-order polar harmonic transforms

Polar harmonic transforms (PHTs) are a kind of continuous orthogonal moments (COMs) that have excellent image description capability. To improve the anti-noise and reconstruction performance of PHTs, integer-order PHTs are generalized to order by a set of novel decimal-order polar harmonic transforms (DPHTs). On this basis, in combination with trinion theory, the DPHTs are extended to trinion decimal-order polar harmonic transforms (TDPHTs) applicable to color images. The color image is processed as a whole, and the internal relations among three components of color images are fully reserved and utilized. Compared with quaternion color image processing, trinion color image processing effectively avoids information redundancy and improves computational efficiency. Color image reconstruction and zero-watermarking algorithm experiments show that TDPHTs have excellent color image description ability and robustness.

A robust zero-watermarking algorithm for lossless copyright protection of medical images

The continuous orthogonal moments of images have become a main tool used in zero-watermarking algorithms in recent years. However, there is a problem that should not be neglected for existing zero-watermarking algorithms, which is known as numerical instability. Because the continuous orthogonal moments used in existing zero-watermarking algorithms are integer-order moments, this problem affects the computational accuracy of such moments and thus affects the performance of zero-watermarking algorithms. Therefore, in this paper, the integer-order radial harmonic Fourier moments (IoRHFMs) are firstly extended to fractional-order radial harmonic Fourier moments (FoRHFMs) to effectively mitigate the problem of numerical instability and improve the computational accuracy of IoRHFMs, and then a FoRHFM-based zero-watermarking algorithm for medical images is proposed to achieve lossless copyright protection of medical images. The experimental results show that this algorithm is highly robust to geometric attacks and common attacks, and is superior to IoRHFM-based and other zero-watermarking algorithms.

Novel fractional-order Jacobi moments and invariant moments for pattern recognition applications

In this paper, we propose a new set of fractional-order continuous orthogonal moments for image representation. These are called fractional-order Jacobi moments (FrJMs) and are defined from the fractional-order orthogonal Jacobi polynomials. We also propose a method for the fast and precise calculation of FrJMs based on recursive calculations of fractional-order Jacobi polynomials and on the separability property of FrJMs. Then, we will derive invariants of FrJMs with respect to rotation, scale and translation (RST) in order to apply them for classification tasks. Just as important, we have presented a systematic parameter selection method for finding the optimal fractional parameter values with respect to pattern recognition applications. Finally, an experimental and comparative study was carried out to test the capacity of FrJMs for the image reconstruction, extraction of global and local image characteristics, invariance to RST, sensitivity to noise, ability to recognize similar grayscale images and the computational times of the new descriptors. The proposed descriptors outperformed the recent orthogonal moments with fractional orders.

Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments.

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

A fast and accurate computation of 2D and 3D generalized Laguerre moments for images analysis

In this paper, we will present a new set of 2D and 3D continuous orthogonal moments based on generalized Laguerre orthogonal polynomials (GLPs) for 2D and 3D image analysis. However, the computation of the generalized Laguerre orthogonal moments (GLMs) is limited by the problems of discretization of the continuous space of the polynomials, approximation of the integrals by finite sums and of too high computation time. To remedy these problems, we will propose a new method for the fast and the precise computation of 2D and 3D GLMs. This method is based on the development of an exact calculation of the double and triple integrals which define the 2D and 3D GLMs, and on the matrix calculation to accelerate the processing time of the images instead of the direct calculation. In addition to the theoretical results obtained, several experiments are carried out to validate the efficiency of the 2D and 3D GLMs descriptors in terms of computation precision and accuracy and in terms of acceleration of computation time and 2D/3D image reconstruction. The experimental results clearly show the advantages and the effectiveness of GLMs compared to the continuous orthogonal moments of Legendre, Chebyshev, Gegenbauer and Gaussian-Hermite.

Image Description With Polar Harmonic Fourier Moments

Due to their good rotational invariance and stability, image continuous orthogonal moments are intensively applied in rotationally invariant recognition and image processing. However, most moments produce numerical instability, which impacts the image reconstruction and recognition performance. In this paper, a new set of invariant continuous orthogonal moments, polar harmonic Fourier moments (PHFMs), free of numerical instability is designed. The radial basis functions (RBFs) of the PHFMs are much simpler than those of the Chebyshev-Fourier moments (CHFMs), orthogonal Fourier-Mellin moments (OFMMs), Zernike moments (ZMs), and pseudo-Zernike moments (PZMs). For the same degree, the RBFs of the PHFMs have more zeros and are more evenly distributed than those of the ZMs and PZMs. Therefore, PHFMs do not suffer from information suppression problem; hence, the image description ability of the PHFMs is superior to that of the ZMs and PZMs. Moreover, the RBFs of the PHFMs are always less than or equal to 1.0 near the unit disk center, whereas those of the OFMMs, PZMs, CHFMs, and radial harmonic Fourier moments (RHFMs) are infinite (implying numerical instability). This indicates that PHFMs can outperform these moments in image reconstruction tasks. We theoretically and experimentally demonstrate that PHFMs outperform the above moments in reconstructing images and recognizing rotationally invariant objects considering noise and various attacks. This paper also details the significance of the PHFM phase in image reconstruction, angle estimation using PHFMs, and the accurate moment selection of the PHFMs.

Comprehensive Study of Continuous Orthogonal Moments—A Systematic Review

Orthogonal moments provide an efficient mathematical framework for computer vision, image analysis, and pattern recognition. They are derived from the polynomials that are relatively perpendicular to each other. Orthogonal moments are more efficient than non-orthogonal moments for image representation with minimum attribute redundancy, robustness to noise, invariance to rotation, translation, and scaling. Orthogonal moments can be both continuous and discrete. Prominent continuous moments are Zernike, Pseudo-Zernike, Legendre, and Gaussian-Hermite. This article provides a comprehensive and comparative review for continuous orthogonal moments along with their applications.

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.

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.

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.

Comparative Study of Exact Continuous Orthogonal Moments Applications : Local Feature Extraction and Data Compression

This paper present an improved reconstruction algorithm of the multi-gray level images based on overlapping block method using exact continuous moments computation: Legendre , Zernike, Pseudo-Zernike and Gegenbauer moments . We solve the artifact issue caused by unitary block reconstruction which affects the visual image quality. This method aim to ensure high accuracy and low computation time, using only small finite number of moments. Our approaches aims to introduce these moments in the field of data compression and local feature extraction for pattern recognition. Experimental results show the superiority of our proposed approaches over the existing methods.

Zernike Moments-Based Gurumukhi Character Recognition

ABSTRACTIn this article, use of Zernike moments is presented for invariant recognition of Gurumukhi characters. Zernike moments belong to a class of continuous orthogonal moments defined over a unit circle. So, for a square image, computations of Zernike moments involve a certain square-to-circle mapping to map the pixel coordinates within the range of the unit circle. There exist two forms of mapping, which are used by various authors in their research works. Here, a computational framework has been proposed for calculation of Zernike moments using both mapping techniques, and a comparison between the performances of both of these mapping techniques is shown through a series of extensive experiments.

Image analysis by generalized Chebyshev–Fourier and generalized pseudo-Jacobi–Fourier moments

In this paper, we present two new sets, named the generalized Chebyshev–Fourier radial polynomials and the generalized pseudo Jacobi–Fourier radial polynomials, which are orthogonal over the unit circle. These generalized radial polynomials are then scaled to define two new types of continuous orthogonal moments, which are invariant to rotation. The classical Chebyshev–Fourier and pseudo Jacobi–Fourier moments are the particular cases of the proposed moments with parameter α=0. The relationships among the proposed two generalized radial polynomials and Jacobi polynomials, shift Jacobi polynomials, and the hypergeometric functions are derived in detail, and some interesting properties are discussed. Two recursive methods are developed for computing radial polynomials so that it is possible to improve computation speed and to avoid numerical instability. Simulation results are provided to validate the proposed moment functions and to compare their performance with previous works.

A Fast Computation of Novel Set of Meixner Invariant Moments for Image Analysis

Recently, the discrete orthogonal moments have been introduced in image analysis and pattern recognition. They have better capacities than the continuous orthogonal moments but the use of these moments as feature descriptors is limited by their high computational cost. As a solution to this problem, we present in this paper an approach for fast computation of Meixner's discrete orthogonal moments. By using the recurrence relation with respect to variable x instead of order n in computation of Meixner's discrete orthogonal polynomials and the image block representation for binary images and intensity slice representation for gray-scale images. The acceleration of the computation time of Meixner moments is due to an innovative image representation where the image is described by a number of homogenous rectangular blocks instead of individual pixels. A novel set of invariant moments based on the Meixner discrete orthogonal moments is also proposed. These invariant moments are derived algebraically from the geometric invariant momenst and their computation is accelerated using image representation scheme. The presented algorithms are tested in several well-known computer vision datasets, regarding computational time, image reconstruction, invariability, and classification. The performances of Meixner invariant moments used as pattern features for a pattern classification application are compared with Hu, Tchebichef, dual-Hahn, and Krawtchouk invariant moments.