Orthogonal Fourier-Mellin Moments Research Articles

Color Image Zero-Watermarking Using Accurate Quaternion Generalized Orthogonal Fourier–Mellin Moments

In recent years, digital image zero-watermarking algorithm has made great progress. But there are still some problems. First, most algorithms only focus on robustness and ignore discrimination. Second, most methods have good robustness against conventional signal attacks but poor robustness against geometric attacks. Thirdly, most of the existing zero-watermarking algorithms only focus on gray-scale images, and there are relatively few researches on color images. In order to cope with these issues, this paper introduces a novel color image zero-watermarking scheme using accurate quaternion generalized orthogonal Fourier–Mellin moments (AQGOFMMs). In the first stage of the proposed method, accurate computation of generalized orthogonal Fourier–Mellin moments (GOFMMs) based on the polar pixel tiling scheme. In the next stage, the high-precision GOFMMs are extended to the accurate quaternion GOFMMs (AQGOFMMs). Finally, the immutable set of features is extracted to construct a zero-watermark. It is worth mentioning that, different from the traditional way of constructing watermark with the amplitude of moments, this algorithm uses the full 4-D features of AQGOFMMs to construct watermark. The experiment proved that proposed zero-watermarking scheme gives a good balance between discriminability and robustness. Furthermore, the proposed algorithm achieves better performance than the other existing zero-watermarking.

Detection of brain tumour by using moments and transforms on segmented magnetic resonance brain images

Brain tumour occurs when abnormal cells appear within the brain. Primary tumour starts with abnormal growth of brain cells whereas Secondary (Metastatic) tumour initiates as cancer in other parts of the body and spread to the brain through blood stream. In this paper, we propose a novel approach to detect tumour in magnetic resonance (MR) brain images. The proposed method uses improved incremental self organise mapping (I2SOM) to segment the brain image and to calculate asymmetry Zernike moments (ZMs), Pseudo-Zernike moments (PZMs) and orthogonal Fourier Mellin moments (OFMMs) are used. It omits the limitation of pre-determination of class of input data and the manual setting of appropriate threshold value. The effectiveness of the proposed method is analysed by doing experiments on 30 MR brain images with tumour and 30 normal MR brain images. It is observed that tumour detection is successfully realised for 30 MR brain images with tumour.

An Efficient Retrieval System for Biomedical Images Based on Radial Associated Laguerre Moments

The ability of any retrieval system to extract features by using its feature descriptor is the primary criterion to measure its efficiency. In this paper a novel technique for feature extraction of biomedical images is presented. The mooted system uses the Radial Associated Laguerre Moments (RALMs) as a feature descriptor to obtain features from two types of medical images: computer tomography (CT) and magnetic resonance images (MRI). RALMs represent one sort of discrete orthogonal moments. RALMs extract the features from images using orthogonal moments to retrieve images from a database. Our approach is extensively assessed with noise-free and noisy images from three different benchmark databases: Emphysema-CT, NEMA CT, and NEMA MRI. The first two databases are used for CT image retrieval, while the third is for MR image retrieval. The proposed approach was tested against the state-of-the art local feature descriptors: Local Binary Pattern (LBP), and local diagonal extrema pattern (LDEP). It was also evaluated against orthogonal Fourier-Mellin moments (OFMMs) as a global descriptor. The comparison shows a significant improvement in favor of the proposed approach in terms of three different performance metrics: ARP, ARR, and F_score. The proposed approach was also compared against the convolutional neural network (CNN) as a deep learning based method over the NEMA-MRI dataset. The RALMs based approach showed a significant improvement when compared against two state-of-the-art medical image retrieval approaches: Histogram of Compressed Scattering Coefficients (HCSCs) and a local bit-plane decoding-based AlexNet descriptor (LBpDAD), the study has done over the TCIA-CT dataset. The proposed approach was also tested with big well-known dataset from the international skin imaging collaboration (ISIC) 2018.

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.

A new approach for effective retrieval and indexing of medical images

The use of medical imaging has increased manifold in past decade. The medical images have become more and more informative about the patient’s anatomy. Thus, the medical images are now not only used for diagnosis purpose but also stored for R&D purpose in order to have better understanding and deeper insights into the cause and cure of various diseases. For real-time retrieval of medical images from such storage repositories there is a grave need of an effective and efficient biomedical image indexing and retrieval approach. In this quest, this paper presents a new approach for the retrieval of CT and MR images using orthogonal Fourier-Mellin moments (OFMMs). OFMMs have excellent information representation capability that enables them to pack the entire image information in very less number of coefficients. This property makes the proposed approach not only effective but also computationally very efficient and most favorable among all the existing approaches. The proposed approach has been tested and compared with numerous existing, state-of-the-art as well as recently published biomedical indexing and retrieval approaches on four standard databases namely, Emphysema CT, NEMA CT, NEMA MRI and OASIS MRI. Additional experiments have been conducted to analyze the noise robustness ability of the proposed and all the compared approaches. The reported results show a significant increase of about 7% and 20% (average) in the retrieval rate of the proposed approach over all the existing approaches on noise free and noisy images, respectively, of all four test medical databases.

Content-based retrieval of biomedical images using orthogonal Fourier-Mellin moments

Biomedical imaging field is growing enormously from last decade. The medical images have been used and stored continuously for diagnosis as well as research purposes. For real-time retrieval of medical images from such storage repositories, there is a grave need of an effective and efficient biomedical image indexing and retrieval approach. In this quest, this paper presents a new approach for the retrieval of CT and MR images using orthogonal Fourier-Mellin moments (OFMMs). OFMMs have excellent information representation capability that enables them to pack the entire image information in very less number of coefficients. This property makes the proposed approach not only effective but also computationally very efficient and most favourable among all the existing approaches. The proposed approach has been tested and compared with numerous existing, state-of-the-art as well as recently published biomedical indexing and retrieval approaches on two standard databases namely, NEMA CT and NEMA MRI. Additional experiments have been conducted to analyse the noise robustness ability of the proposed and all the compared approaches. The reported results show superior retrieval performance of the proposed approach and a significant increase in the retrieval rate over all the existing approaches on noisy images on both the test databases.

Quaternion polar harmonic Fourier moments for color images

This paper proposes quaternion polar harmonic Fourier moments (QPHFM) for color image processing and analyzes the properties of QPHFM. After extending Chebyshev–Fourier moments (CHFM) to quaternion Chebyshev-Fourier moments (QCHFM), comparison experiments, including image reconstruction and color image object recognition, on the performance of QPHFM and quaternion Zernike moments (QZM), quaternion pseudo-Zernike moments (QPZM), quaternion orthogonal Fourier-Mellin moments (QOFMM), QCHFM, and quaternion radial harmonic Fourier moments (QRHFM) are carried out. Experimental results show QPHFM can achieve an ideal performance in image reconstruction and invariant object recognition in noise-free and noisy conditions. In addition, this paper discusses the importance of phase information of quaternion orthogonal moments in image reconstruction.

Constant-Time Calculation of Zernike Moments for Detection with Rotational Invariance

We construct a set of special complex-valued integral images and an algorithm that allows to calculate Zernike moments fast, namely in constant time. The technique is suitable for dense detection procedures, where the image is scanned by a sliding window at multiple scales, and where rotational invariance is required at the level of each window. We assume no preliminary image segmentation. Owing to the proposed integral images and binomial expansions, the extraction of each feature does not depend on the number of pixels in the window and thereby is an O(1) calculation. We analyze algorithmic properties of the proposition, such as: number of needed integral images, complex-conjugacy of integral images, number of operations involved in feature extraction, speed-up possibilities based on lookup tables. We also point out connections between Zernike and orthogonal Fourier-Mellin moments in the context of computations backed with integral images. Finally, we demonstrate three examples of detection tasks of varying difficulty. Detectors are trained on the proposed features by the RealBoost algorithm. When learning, the classifiers get acquainted only with examples of target objects in their upright position or rotated within a limited range. At the testing stage, generalization onto the full 360° angle takes place automatically.

An efficient and robust multi-frame image super-resolution reconstruction using orthogonal Fourier-Mellin moments

Multi-frame image super-resolution (SR) has recently become an active area of research. The orthogonal rotation invariant moments (ORIMs) have several useful characteristics which make them very suitable for multi-frame image super-resolution application. Among the various ORIMs, Zernike moments (ZMs) and pseudo-Zernike moments (PZMs)-based SR approaches, i.e., NLM-ZMs and NLM-PZMs, have already shown improved SR performances for multi-frame image super-resolution. However, it is a well-known fact that among many ORIMs, orthogonal Fourier-Mellin moments (OFMMs) demonstrate better noise robustness and image representation capabilities for small images as compared to ZMs and PZMs. Therefore, in this paper, we propose a multi-frame image super-resolution approach using OFMMs. The proposed approach is based on the NLM framework because of its inherent capability of estimating motion implicitly. We have referred to this proposed approach as NLM-OFMMs-I. Also, a novel idea of using OFMMs-based interpolation in place of traditional Lanczos interpolation for obtaining an initial estimate of HR sequence has been presented in this paper. This variant of the proposed approach is referred to as NLM-OFMMs-II. Detailed experimental analysis demonstrates the effectiveness of the proposed OFMMs-based SR approaches to generate high-quality HR images in the presence of factors like image noise, global motion, local motion, and rotation in between the image frames.

Single-image super-resolution using orthogonal rotation invariant moments

In this paper, we propose an interpolation-based single-frame image super-resolution approach using orthogonal rotation invariant moments (ORIMs). The ORIMs have several useful characteristics in addition to having the property of image reconstruction. Therefore, they have been used successfully in many image processing applications. Among the various ORIMs, Zernike moments (ZMs), pseudo-Zernike moments (PZMs) and orthogonal Fourier-Mellin moments (OFMMs) have been considered in our proposed framework. The SR performance of the proposed approach has been compared with the classical interpolation-based approaches like bicubic, cubic B-spline, and Lanczos, as well as with nonlocal-means (NLM), and recently developed NLM+ZMs and NLM+PZMs-based SR approaches on twelve standard test images. The experiments have been conducted on both noise-free and noisy LR images corrupted with uniform blur and Gaussian noise. The results demonstrate the superiority of the proposed ORIMs-based approach in super-resolving both noise-free and noisy images over recently developed NLM+ORIMs-based SR approaches. Also, a comparative performance analysis, among various ORIMs (ZMs, PZMs, and OFMMs), is also presented to determine the ORIM which performs better over others under a given condition. A time complexity analysis shows that the proposed method is very fast as compared to NLM, NLM+ZMs and NLM+PZMs-based methods.

Rotation, scaling and translation invariant texture recognition by Bessel-Fourier moments

The ideal of Bessel-Fourier moments (BFMs) for image analysis and only rotation invariant image cognition has been proposed recently. In this paper, we extend the previous work and propose a new method for rotation, scaling and translation (RST) invariant texture recognition using Bessel-Fourier moments. Compared with the others moments based methods, the radial polynomials of Bessel-Fourier moments have more zeros and these zeros are more evenly distributed. It makes Bessel-Fourier moments more suitable for invariant texture recognition as a generalization of orthogonal complex moments. In the experiment part, we got three testing sets of 16, 24 and 54 texture images by way of translating, rotating and scaling them separately. The correct classification percentages (CCPs) are compared with that of orthogonal Fourier-Mellin moments and Zernike moments based methods in both noise-free and noisy condition. Experimental results validate the conclusion of theoretical derivation: BFM performs better in recognition capability and noise robustness in terms of RST texture recognition under both noise-free and noisy condition when compared with orthogonal Fourier-Mellin moments and Zernike moments based methods.

Robust watermarking using orthogonal Fourier–Mellin moments and chaotic map for double images

In this paper, a robust watermarking scheme based on orthogonal Fourier–Mellin moments and chaotic map is introduced, which achieves copyright authentication for double images simultaneously. The proposed scheme consists of an ownership registration phase and a verification phase. Compared to the congeneric schemes, the new scheme achieves a high level security and saves storage space. Firstly, two images are combined into a single-channel architecture, then the feature invariants derived from orthogonal Fourier–Mellin moments are computed and utilized to construct a binary feature image. Together with the watermark, they are scrambled by chaotic map and subsequently used for generating the verification image. Experimental results demonstrate the validity and security of the proposed scheme as well as its robustness against different attacks.

Image analysis by circularly semi-orthogonal moments

Various types of circularly orthogonal moments have been widely used for image reconstruction and rotation invariant classification. However, they suffer from two errors namely numerical integration error and geometric error, which affect their reconstruction capability and pattern recognition accuracy. In this paper, a novel category of circular moments named circularly semi-orthogonal moments is proposed. In the proposed moment, a set of orthogonal basis functions modulated by a negative power exponential envelope is utilized as the radial basis function. For a given degree n, the radial basis function possesses more compact bandwidth, less cutoff frequency and more zeros compared with the frequently-used circularly orthogonal moments including Zernike and orthogonal Fourier–Mellin moments, and so the circularly semi-orthogonal moment calculated with the zeroth order approximation is more robust to numerical error than the frequently-used circularly orthogonal moments. Furthermore, the capability of the semi-orthogonal moment to describe high spatial frequency components of images is relatively higher than that of the frequently-used circularly orthogonal moments. Experimental results demonstrate that the semi-orthogonal moments calculated with the zeroth order approximation perform better than the frequently-used circularly orthogonal moments in terms of image reconstruction capability and invariant recognition accuracy in noise-free, noisy and smooth distortion conditions. It is also shown that the proposed high order moments are more numerically stable than the circularly orthogonal moments.

General form for obtaining unit disc-based generalized orthogonal moments.

The rotation invariance of the classical disc-based moments, such as Zernike moments (ZMs), pseudo-ZMs (PZMs), and orthogonal Fourier-Mellin moments (OFMMs), makes them attractive as descriptors for the purpose of recognition tasks. However, less work has been performed for the generalization of these moment functions. In this paper, four general forms are developed to obtain a class of disc-based generalized radial polynomials that are orthogonal over the unit circle. These radial polynomials are scaled to ensure numerical stability, and some useful properties are discussed for potential applications they could be used in. Then, these scaled radial polynomials are used as kernel functions to construct a series of unit discbased generalized orthogonal moments (DGMs). The variation of parameters in DGMs can form various types of orthogonal moments: 1) generalized ZMs; 2) generalized PZMs; and 3) generalized OFMMs. The classical ZMs, PZMs, and OFMMs correspond to a special case of these three generalized moments for which the free parameter α = 0. Each member of this family will share some excellent properties for image representation and recognition tasks, such as orthogonality and rotation invariance. In addition, we have also developed two algorithms, the so-called m-recursive and n-recursive methods for the computation of these proposed radial polynomials to improve the numerical stability. Experimental results show that the proposed methods are superior to the classical disc-based moments in terms of image representation capability and classification accuracy.

Error Analysis in the Computation of Orthogonal Rotation Invariant Moments

Orthogonal rotation invariant moments (ORIMs) are among the best region based shape descriptors. Being orthogonal and complete, they possess minimum information redundancy. The magnitude of moments is invariant to rotation and reflection and with some geometric transformation, they can be made translation and scale invariant. Apart from these characteristics, they are robust to image noise. These characteristics of ORIMs make them suitable for many pattern recognition and image processing applications. Despite these characteristics, the ORIMs suffer from many digitization errors, thus they are incapable of representing subtle details in image, especially at high orders of moments. Among the various errors, the image discretization error, geometric and numerical integration errors are the most prominent ones. This paper investigates the contribution and effects of these errors on the characteristics of ORIMs and performs a comparative analysis of these errors on the accurate computation of the three major ORIMs: Zernike moments (ZMs), Pseudo Zernike moments (PZMs) and orthogonal Fourier-Mellin moments (OFMMs). Detailed experimental analysis reveals some interesting results on the performance of these moments.

Quaternion Bessel–Fourier moments and their invariant descriptors for object reconstruction and recognition

In this paper, the quaternion Bessel–Fourier moments are introduced. The significance of phase information in quaternion Bessel–Fourier moments is investigated and an accurate estimation method for rotation angle is described. Furthermore, a new set of invariant descriptors based on the magnitude and the phase information of quaternion Bessel–Fourier moments is derived. Experimental results show that quaternion Bessel–Fourier moments lead to better performance for color image reconstruction than the other quaternion orthogonal moments such as quaternion Zernike moments, quaternion pseudo-Zernike moments and quaternion orthogonal Fourier–Mellin moments. In addition, the angles estimated by the proposed moments are more accurate than those obtained by using other quaternion orthogonal moments. The proposed invariant descriptors show also better robustness to geometric and photometric transformations.

Generic radial orthogonal moment invariants for invariant image recognition

As the variation of parameters in Jacobi polynomial, Jacobi–Fourier moments can form various types of orthogonal moments: Legendre–Fourier moments, Orthogonal Fourier–Mellin moments, Zernike moments, pseudo-Zernike moments, and so on. In this paper, we present a generic approach based on Jacobi–Fourier moments for scale and rotation invariant analysis of radial orthogonal moments, named Jacobi–Fourier moment invariants (JFMIs). It provides a fundamental mathematical tool for invariant analysis of the radial orthogonal moments since Jacobi–Fourier moments are the generic expressions of radial orthogonal moments. Theoretical and experimental results also show the superiority of the proposed method and its robustness to noise in comparison with some exist methods.

Accurate Computation of Orthogonal Fourier-Mellin Moments

Orthogonal Fourier-Mellin moments (OFMMs) suffer from geometric error and the numerical integration error. The geometric error arises when the square image is mapped into a unit disk and the mapping does not become perfect. The numerical integration error arises when the double integration is approximated by the zeroth order summation. In this paper, we propose methods which reduce these errors. The geometric error is reduced by considering the arc-grids lying on the boundary of the unit disk and the square grids lying completely inside the disk. The numerical integration error is reduced by Gaussian numerical integration, for which a simple computational framework is provided. The relative contributions of geometric error and numerical integration error to the total error are also analyzed. It is observed that the geometric error is significant only for the small images whereas the magnitude of numerical integration is significantly high for all image sizes, which increases with the order of moments. A simple computational framework which is similar to the conventional zeroth order approximation is also proposed which not only reduces numerical integration error but also reduces geometric error without considering arc-grids. The improved accuracy of OFMMs are shown to provide better image reconstruction, numerical stability and rotation and scale invariance. Exhaustive experimental results on a variety of real images have shown the efficacy of the proposed methods.

Local and global features based image retrieval system using orthogonal radial moments

Orthogonal radial moments such as Zernike moments, pseudo Zernike moments, and orthogonal Fourier Mellin moments have been studied extensively in the literature. In conventional methods of moment computation, the entire image intensity function is projected upon orthogonal moment polynomials to compute the global features. In this paper, we provide a novel methodology for the computation of these moments, which formulate them as effective local descriptors. The moment functions of edge images are computed to determine the local change in images, which provide the local aspects of the image. Thus, the obtained local and global moment features are combined to evaluate their performance on the image retrieval system. The moments ZMs, PZMs, and OFMMs, are compared in terms of their image retrieval effectiveness. The experiments are performed on various databases to examine the performance of the system in diverse circumstances such as images affected with noise, partial occlusion, distortion, complex structure, etc. The experiments results reveal that the proposed system outperforms the existing recent approaches to moments computation for image retrieval.

On image reconstruction, numerical stability, and invariance of orthogonal radial moments and radial harmonic transforms

Various radial moments viz. Zernike moments, pseudo Zernike moments, orthogonal Fourier Mellin moments, radial harmonic Fourier moments, Chebyshev-Fourier moments and polar harmonic transforms such as polar complex exponential transforms, polar cosine transforms and polar sine transforms satisfy orthogonal principle. By virtue of which these moments and transforms possess minimum information redundancy and thereby exhibit a good characteristic of image representation. In this paper, a complete comparative analysis is performed by considering image reconstruction capability of each individual moment and transform. The orthogonal properties of above mentioned moments along with the causes of their, reconstruction error, numerical stability and invariance are described.