An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments

This paper introduces an efficient method for face localization and recognition in color images. The proposed method uses the location of eyes for computation and extraction of a face's bounding ellipse. In this way, parameters of a face's ellipse (center, orientation, major and minor axis), is computed by the location of eyes in a face image. In the next step, we apply Pseudo Zernike Moments (PZM), Zernike Moments (ZM) and Principal Component Analysis (PCA) for feature extraction. For classification of these feature vectors a new structure of RBF neural networks with a novel distance function is introduced and a new method for determination of RBF unit parameters is proposed. Finally, we compare the efficiency of the proposed system for three types of feature vectors (PZM, ZM and PCA). Results emphasize the high accuracy and efficiency of the PZM features proportion to other features (ZM and PCA) for use in the proposed recognition system.

Analysis of noise sensitivity of Tchebichef and Zernike moments with application to image watermarking

Discrete Chebyshev moments (due to discrete polynomial basis) do not have the discretization errors that continuous-domain Legendre and Zernike moments contain. Calculation of polynomial basis coefficients of discrete moments is generally performed using recurrence relationships. Such recurrence equations cause numerical error accumulation especially for calculation of higher-order moments and for larger image sizes, causing significant degradation of image reconstruction from these moments. A method for better image reconstruction from high orders of discrete Chebyshev moments is demonstrated. This is accomplished by calculating Chebyshev polynomial coefficients directly from their definition formulas using arbitrary precision arithmetic and by forming lookup tables from these coefficients.

The fast computation of Zernike moments from normalized geometric moments has been developed in this paper. The computation is multiplication free and only additions are needed to generate Zernike moments. Geometric moments are generated using Hatamian's filter up to high orders by a very simple and straightforward computation scheme. Other kinds of moments (e.g., Legendre, pseudo Zernike) can be computed using the same algorithm after giving the proper transformations that state their relations to geometric moments. Proper normalizations of geometric moments are necessary so that the method can be used in the efficient computation of Zernike moments. To ensure fair comparisons, recursive algorithms are used to generate Zernike polynomials and other coefficients. The computational complexity model and test programs show that the speed-up factor of the proposed algorithm is superior with respect to other fast and/or direct computations. It perhaps is the first time that Zernike moments can be computed in real time rates, which encourages the use of Zernike moment features in different image retrieval systems that support huge databases such as the XM experimental model stated for the MPEG-7 experimental core. It is concluded that choosing direct computation would be impractical.

The moment invariants have been widely used as feature descriptors against both blur degradation and similarity transformation in pattern recognition. Since Pseudo-Zernike moments outperform other moments in feature matching and object recognition, we propose a new set of Pseudo-Zernike moment invariants holding for blur and similarity transformation simultaneously in this paper to improve the match results. In order to construct the proposed invariants, we also establish the relationship between the Pseudo-Zernike moments of the distorted images and those of the original images with respect to circularly symmetric point spread function. The experimental results are presented to confirm that compared to the other moment invariants, the proposed combined invariants have better invariance and more powerful discriminating ability in terms of pattern recognition accuracy, especially when the images contain noises.

We present invariant image watermarking based on a recently introduced set of polar harmonic transforms and angular radial transforms and their comparative analysis with state-of-art approaches based on Zernike moments and pseudo-Zernike moments (ZMs/PZMs). Similar to ZMs/PZMs, these transforms provide rotation invariance and resilience to noise while mitigating inherent limitations like numerical instability and computational cost at high order of moments. These characteristics motivate us to design invariant transform-based invariant image watermarking schemes that can withstand various intentional or unintentional attacks, handle large bitcarriers, and work in a limited computing environment. A comparative performance evaluation of watermarking systems regarding critical parameters like visual imperceptibility, embedding capacity, and watermark robustness against geometric transformations, common signal processing distortions, and Stirmark attacks is performed along with the empirical analysis of various inherent properties of transforms and moments such as magnitude invariance, reconstruction capabilities, and computational complexity to investigate relationships between the performance of watermarking schemes and inherent properties of transforms.

This paper presents the use of Hidden Markov Models (HMM) to classifying Arabic words that have been represented by moments into a set of predefined classes. The proposed scheme results in enhanced performance because it deals effectively with errors that commonly originate during the feature extraction stage and it accounts for variations due to the individual handwriting performance of different data sets. To overcome the variations in images, each is represented by rotation and scale invariant feature vector known as Pseudo Zernike Moments (PZM). To demonstrate its effectiveness, the proposed classification scheme has been employed in the context of different sets of handwritten Arabic words. The classification results demonstrate that the proposed approach can classify handwritten Arabic word with a error rate less than 10%.

Simple and fast computation of moments

As an orthogonal moment, Zernike moment (ZM) is an attractive image feature in a number of application scenarios due to its distinguishing properties. However, we find that for digital images, the commonly used Cartesian method for ZM computation has compromised the advantages of ZMs because of their non-ideal accuracy stemming from two inherent sources of errors, i.e., the geometric error and the integral error. There exists considerable errors in image reconstruction using ZMs calculated with the Cartesian method. In this paper, we propose a polar coordinate based algorithm for the computation of ZMs, which avoids the two kinds of errors and greatly improves the accuracy of ZM computation. We present solutions to the key issues in ZM computation under polar coordinate system, including the derivation of computation formulas, the polar pixel arrangement scheme, and the interpolation-based image conversion etc. As a result, ZM-based image reconstruction can be performed much more accurately.

One of the main contributions of this paper is to show that contrary to the common belief Tchebichef moments are more sensitive to noise added to the image than Zernike moments. We show this by imposing different types and levels of noise on various images and measuring the error due to the added noise alone after image reconstruction. To calculate the error due to added noise alone, the image reconstruction error is subtracted from the total error, which initially includes both reconstruction and added noise errors. Another contribution of this paper is to present some analysis and compare the reconstruction capability of Tchebichef moments to that of Zernike and Legendre moments in noise free environments independent of the image used. We show this by spectrally analyzing the performance of Tchebichef, Legendre, and Zernike moments using single frequency images. Here the total reconstruction error of an arbitrary image is equal to the integration of the spectral errors, whose transfer functions are calculated using single frequency images, for all frequencies.

Quaternion polar complex exponential transform (QPCET) moments and their invariants are widely used as powerful tools in many image processing and pattern recognition applications. However, the accuracy of the conventional approximated method for computing QPCET moments suffers from geometric and numerical errors. This approximated method is very time-consuming. Moreover, computing the high orders of approximated QPCET moments suffers from numerical instability. Computational methods are proposed for fast and accurate computation of the QPCET moments for color images in two coordinate systems. In the first method, the Gaussian quadrature method is applied to compute higher-order moments of QPCET in the Cartesian coordinates. On the other side, an exact kernel-based method is employed to compute the higher-order moments of QPCET in the polar coordinates. A set of numerical experiments is conducted and the obtained results clearly show that the conventional approximated method is unstable, where the numerical instability encountered with moment order ≥10, while the first proposed method is unstable for moment order ≥60. On the other side, the second proposed method is stable for all orders. The comparison clearly shows the superiority of the second proposed method in terms of image reconstruction capability, numerical stability, fast computation, rotation invariances, and robustness to different kinds of noises.

Assessment of cluster analysis of elastic light scatter profiles for the identification of foodborne Bacteria.

Image reconstruction is very much concerned with the ability to reproduce image of possibly the same quality as original. This paper presents the comparison of Zernike Moment (ZM) and Discrete Cosine Transform (DCT) for image reconstruction in noise environment. Two types of images used are gray scale and binary. The original images are corrupted with three different noises that are Salt and Pepper, Gaussian and Random. Both original and corrupted images are reconstructed using Inverse Zernike Moment and Inverse Discrete Cosine Transform. The performance of each algorithm is measured by evaluating Peak Signal to Noise Ratio (PSNR) for the reconstructed gray scale of pixel size 64×64 and binary of pixel size 30×30 images. The comparison of PSNR values between the two techniques proves that Zernike Moment is less sensitive to noisy images compared to Discrete Cosine Transform.

Invariant image representation plays an important role in many pattern recognition applications, such as texture classification, face recognition and character recognition. In this paper, we evaluate some of the invariant orthogonal moments, including Zernike moment (ZM), pseudo-Zernike moment (PZM), and Polar Harmonic Transform (PHT), which are all computed in a circular domain. The performance of invariance and image representation ability are evaluated.

A novel method for early diagnosis of Alzheimer’s disease based on pseudo Zernike moment from structural MRI