Image analysis with logarithmic Zernike 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.

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

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.

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.

Zernike moments (ZMs) are widely used in many image analysis and pattern recognition problems because of their superiority compared with other moments. However, they suffer from high computation cost and inherent error. Previous researches have shown that the algorithm, computing ZMs in polar system, improves the ZMs accuracy of the reconstruction and invariance properties dramatically. In this study, the authors firstly modify a direct method for computing ZMs in polar coordinates and present a recursive relation. Then, this study presents an algorithm for fast computation of ZMs, based on the improved polar pixel tiling scheme. Owing to the symmetrical property, ZMs can be obtained by computing only one-sixteenth circle of the radial polynomials, which means that the number of pixels involved in the computation of ZMs is only 6.25% of the previous method. This leads to a significant reduction in the computational complexity requirements. A comparison with the other conventional method is performed in detail. The obtained results show the superiority of the proposed method.

This paper proposes an approach combining the set of Krawtchouk moments and HMMs (Hidden Markov Models) for unconstrained offline handwritten Chinese character recognition. The Krawtchouk moments can be effectively used as pattern features in analysis of two-dimensional images. Since the basis set is orthogonal in the discrete domain of the image coordinate space, the implementation of Krawtchouk moments doesn't involve any numerical approximation or coordinate transformation. This property makes Krawtchouk moments superior to conventional, more continuous moments such as Legendre moments and Zernike moments. In the field of character recognition, HMMs architecture has been used widely because of its strong adaptability to written distortion and variability. In this paper, Krawtchouk moments and HMMs are combined to recognize unconstrained offline handwritten Chinese amount in words. Experiments show that the average recognition of Krawtchouk is higher than Zernike and Legendre moments within the HMM framework.

Aims/ objectives: To demontrate effectiveness of Zernike Moments for Image Classification. Zernike moment(ZM) is an excellent region-based moment which has attracted the attentions of many image processing researchers since its first application to image analysis. Many papers have been published on several works done on ZM but no single paper ever give a detailed information of how the computation of ZM is done from the time the image is captured to the computation of ZM. This work showed how to effectively apply ZM on RGB images. We have demonstrated the effectiveness of Zernike moment in image classification system. A neuro-genetic intelligent system has been built with PNN classifier. The feature extracted viz ZM and Geometric features were further subjected to GA to bring the best combinatorial features for optimal accuracy. The algebraic structure of our novel fitness function enabled the GA to select the best results. The 10-fold CV used enabled the whole system to be unbiased giving a classification accuracy of 90.05%. A demonstration of affine properties of ZM are comprehensively stated and explained. In summary, the ZM enabled the classifier to have improved accuracy of 91% as compared with Geometric features with 89% accuracy.

Zernike moments (ZMs) are a set of orthogonal moments which have been successfully used in the fields of image processing and pattern recognition. A combination of edge blurring with ZMs computation was introduced. In this study, several kinds of artificial binary stripe images were used to investigate the effects of edge blurring on the absolute mean error of reconstructed image from high-order ZMs. After the blurring process, the reconstruction errors were increased dramatically at edge pixels, but decreased on non-edge pixels. The experimental results demonstrated that 2-pixel blurring approach provided better performance for reducing reconstruction error. Finally, a template matching between two real images was simulated to illustrate the effectiveness of the proposed method.

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.

Zernike Moments (ZMs) have been widely used in pattern recognition and image analysis due to their merits of near-zero information redundancy and rotation invariance. However, the discrete sampling error introduced by state-of-the-art algorithms degrades their rotation invariance. In this paper, a novel algorithm for computing the accurate ZMs of binary images is presented. The new algorithm is based on Non-Symmetry Anti-Packing Image Representation (NAIR) and it computes the accurate ZMs of the whole image by summing up the ZMs of all rectangular regions without introducing the discrete sampling error. The experimental results in this paper show that the NAIR-based algorithm for accurate ZMs is faster than the state-of-the-art accurate algorithms.

Changes in illumination condition, pose, facial expression and others is not an easy task in face recognition. Solving these problems requires not only a feature extraction method that can generate distinct features for each class of image but requires other additional technique that able to improve the overall classification accuracy. This paper presents the face recognition using combined features of original and pre-processed face images. This technique is experimented using orthogonal moments namely Zernike moments (ZMs) and Krawtchouk moments (KMs). The classification technique used in the recognition stage is Euclidean square distance or Nearest Neighbour (NN) classifier. Database face images from Olivetti research laboratory (ORL) consisting of 40 subjects of 10 images each where none of them are identical, is used in the experiments. The face images vary in position, rotation, scale and expression, with and without spectacles. From the experiments, the new technique is able to improve the classification accuracy significantly.

Various types of orthogonal moments have been widely used for object recognition and classification. This paper presents an effective way of extracting texture features, Bessel Fourier moments, for image retrieval and classification applications. The Bessel Fourier moments are calculated for rotation invariance and perform better in terms of represent global features than orthogonal Fourier-Mellin and Zernike moments. In order to achieve good results in CBIR experiments and image classification experiments using Bessel Fourier moments, we conduct experiments on four databases: the first one is a small 2D texture database formed by 400 images, and the rest are 2D color image databases with different relevant images formed by 1200 from the Amsterdam Library of Object Image (ALOI). The experiments show that the feature descriptors extracting with the proposed algorithm perform better for image retrieval and classification than conventional descriptors by comparing the retrieval accuracy, the same order.

Surveillance and security cameras help security forces in public places such as airports, railway stations, universities and office buildings to perform high-level surveillance tasks such as detecting suspicious activity or anticipating undesirable events. Re-Identification (Re-ID) is defined as the process of communicating between images of the person in different cameras in a surveillance environment. Changing the field of view of any camera presents challenges such as changing body posture, changing brightness, noise and blockage. This article focuses on extracting the most distinctive features to overcome these challenges. The features of Hu moment, Zernike moment in 9th order and Legendre moment in 9th order for each image are extracted and merged into a single feature vector to form a single feature vector for each image. Principal Component Analysis (PCA) was used to reduce the vector dimensionality and finally the Mahalanobis distance criterion was used for identification. The proposed method in the VIPeR database has achieved a re-ID rate of 96.5. Although the presented method is simple, the outcome has been superior compared to many of the state-of-the-art methods.

Impact of the Radial Basis Function Spread Factor onto Image Reconstruction in Electrical Impedance Tomography

In this work we present an optical-digital system that uses a liquid crystal display (LCD)for the approximate reconstruction of binary images by using the Zernike moments. We use the fact that, each Zernike polynomial can be mapped as an intensity distribution and therefore can be readily displayed with a LCD in an optical incoherent system. On the other hand, the reconstruction of the input image can be obtained from a discrete sum of the data matrices corresponding to the Zernike images weighted by suitable coefficients. These can be numerically computed by using their orthogonality property. The Zernike images acquired by the optical-digital system generate a basis set for the image reconstruction, which depends on both LCD and CCD sensor spatial resolution. The reconstruction is implemented with only thirty six real-valued Zernike moments.