Explicit invariance of Cartesian Zernike moments

Various moment invariants have been developed for object recognition applications. In this paper, we conduct a comparative study of different moment invariants with respect to perspective transformation. Perspective transformations are induced by the lens of the human eye or an optical system. A physical model of the perspective transformation is given. concepts of Zernike moments and regular moments are discussed and their derivation compared. Their corresponding rotational, scale, and translational invariance properties are presented. A group of quadratic form perspective moment invariants is also introduced. Comparisons and estimations are made on the performance and properties of these invariants in object recognition from both a theoretical and a computational point of view. After the overall estimation, it is shown that the most effective way in recognizing object under perspective transformation is to use quadratic form invariants.

We present a way to construct a complete set of scaling rotation and translation invariants extract directly from Zernike moments. Zernike moment can be constructed by Radial moment. In our method in order to construct invariant Zernike moment is to achieve invariant Radial moment which is component of Zernike moment. We use matrix form to denote relationship between Radial and Zernike moment, which makes derivation more comprehensible. The translation invariant Radial moment is first introduced, for it is most complicated part of all the three invariant. Rotation and scaling invariant Radial moment is achieved by normalizing the factor caused by rotation and scaling. The form of invariant radial moment is to combine three parts of invariant. Some experiment has done to test the performance of invariance. In this experiment we take an image library containing 23,329 files which are built by translation rotation and zoom in out of one origin Latin character image. Most of the value of standard deviation ratio by mean of proposed moments is nearly 1%. In addition, retrieval experiment is to test the discrimination ability. MPEG-7 CE shape1 - Part A library is taken in this experiment. The recall rate in part A1 is 96.6% and is 100% in part A2.

The paper proposes a neural network technique to classify numerals using Zernike moments that are invariant to rotation only. In order to make them invariant to scale and shift, we introduce modified Zernike moments based on regular moments. Owing to the large number of Zernike moments used, it is computationally more efficient to select a subset of them that can discriminate as well as the original set. The subset is determined using stepwise discriminant analysis. The performance of a subset is examined through its comparison to the original set. The results are shown of using such a scheme to classify scaled, rotated, and shifted binary images and images that have been perturbed with random noise. In addition to the neural network approach, the Fisher's classifier is also used, which is a parametric classifier. A comparative study of their performances shows that the neural network approach produces better classification accuracy than the Fisher's classifier. When a suitable subset of Zernike moments is used, the classifiers perform well, just like the original set. The performance of the classifiers is also examined. The computational time is greatly reduced when a suitable subset of Zernike moments is used.

A comprehensive treaty on the theory and applications of moment functions in image analysis. Moment functions are widely-used in various realms of computer vision and image processing. Numerous algorithms and techniques have been developed using image moments, in the areas of pattern recognition, object identification, three-dimensional object pose estimation, robot sensing, image coding and reconstruction. This is a compilation of the theoretical aspects related to different types of moment functions, and their applications in the above areas. The book is organized in two parts. The first part discusses the fundamental concepts behind important moments such as geometric moments, complex moments, Legendre moments, Zernike moments, and moment tensors. Most of the commonly-used properties of moment functions and the mathematical framework for the derivation of basic theorems and results are discussed. This includes the derivation of moment invariants, implementation aspects of moments, transport properties, and fast methods for computing the moment functions for both binary and gray-level images. The second part presents the key application areas of moments, such as pattern recognition, object identification, image-based pose estimation, edge detection, clustering, segmentation, coding and reconstruction. Important algorithms in each of these areas are discussed. A list of bibliographic references on image moments is also included.

Feature descriptors for 3-D images have recently gained considerable attention in application for games, virtual reality environment and solid modeling. Numerous research had been introduced for 3-D invariants of geometric moments, complex moments and Zernike moments. In this paper, we present a theoretical framework to derive translation and scale invariants for 3-D Legendre moments, by using indirect and direct methods. Indirect method generates 3-D Legendre invariants from the existing 3-D geometric moment invariants. Direct method, on the other hand, eliminates the displacement and scale factors from Legendre polynomials to generate translation and scale invariants. Experiment using 3-D binary images are carried out to verify the proposed feature descriptors.

Finite precision error analysis of Zernike moments computation schemes and a new, efficient, robust recursive algorithm

According to the traffic sign recognition process converges slowly, easy to fall into local minimum of defects in general BP algorithm, a BP algorithm is discussed in the paper, which is based on the improved conjugate gradient method on the basis of Fletcher-Reeves linear search method. At the beginning, the basic principle of the algorithm is introduced; meanwhile, an in-depth analysis is discussed from the theoretical aspects in the paper, and then the trained BP neural network is applied into the function approximation. Experimental results show that the algorithm has many advantages, such as it has a good effect in solving the rotation and scale invariance with the image, the extracted features with translation, scale and rotation invariance. The fast convergence speed and less iteration of this algorithm can meet the requirements of unmanned vehicle autonomous navigation.

This paper proposes a method of facial expression recognition based on Zernike moments and the minimum classification error (MCE) based hidden Markov model (HMM). In the feature extraction of face, the method of Zernike moments feature extraction based on local feature regions is adopted. First, eyes and mouths are segmented from the facial expression image and Zernike moments feature vectors of eyes and mouth are extracted. In the classification of expression, using MCE criteria is adopted to improve the performance of HMM. The results show that the performance of expression recognition system can be improved to some extent by using Zernike moments and MCE criterion.

The wind power sector is experiencing rapid growth, which creates new challenges for its electricity grid integration. Accurate wind power forecasting (WPF) is crucial for trading, balancing, and dispatching wind energy. In this paper, we examine the use of aggregated turbine- and farm-level WPFs in the Nordic energy market. The turbine-level WPFs were retrieved from a previous study, while the farm-level WPFs were developed using the same methodology, incorporating inputs from three different numerical weather predictions (NWPs) and implementing both direct and indirect forecasting approaches. In the indirect WPF approach, we explore the impact of using wind direction as an input for the wind farm-level power performance model. The different WPFs are combined into one using weights related to up-to-date forecast errors. An automated and optimized machine-learning pipeline using data from a Norwegian wind farm is used to implement the proposed forecasting methods. The indirect approach, that uses the wind-downscaling model, improves the wind speed forecast accuracy compared to raw forecasts from the relevant NWPs. Additionally, we observed that the farm-level downscaling model exhibited lower error than those developed at the turbine level. The combined use of multiple NWP sources reduced forecasting errors by 8 %–30 % for direct and indirect WPFs, respectively. Direct and indirect forecasting methods present similar performance. Finally, the aggregated turbine-level improved WPF accuracy by 10 % and 15 % for RMSE and MAE, respectively, compared to farm-level WPF.

Ladar range images have attracted considerable attention in automatic target recognition fields. In this paper, Zernike moments (ZMs) are applied to classify the target of the range image from an arbitrary azimuth angle. However, ZMs suffer from high computational costs. To improve the performance of target recognition based on small samples, even-order ZMs with serial-parallel backpropagation neural networks (BPNNs) are applied to recognize the target of the range image. It is found that the rotation invariance and classified performance of the even-order ZMs are both better than for odd-order moments and for moments compressed by principal component analysis. The experimental results demonstrate that combining the even-order ZMs with serial-parallel BPNNs can significantly improve the recognition rate for small samples.

Shape analysis requires invariance under translation, scale, and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modeling the resulting data using complex scalar-rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters and the nonlinear nature of their formulation. In the present paper, we show how a set of kernel functions that we refer to as rotation invariant kernels can be used to convert the original nonlinear problem into a linear one. As their name implies, these kernels are defined to provide the much needed rotation invariance property allowing one to bypass the difficulty of working with complex spherical distributions. The resulting approach provides an easy, fast mechanism for 2D & 3D shape analysis. Extensive validation using a variety of shape modeling and classification problems demonstrates the accuracy of this proposed approach.

Two important factors that determine the efficiency and reliability of a rice sorting machine are the overall processing speed and the classification accuracy. In this paper, an efficient rice sorting process which uses a subset of Zernike moments (ZM) and a multilayer perceptron is presented. Since the falling rice grains during sorting process can be in any orientation, a rotational invariant feature set is crucial in this application. Hence, the set of ZM with its inherent rotational invariance property is chosen in this context. Nevertheless, one of the main drawbacks of ZM in real-time application is its long computation time. To overcome this, a subset of ZM is selected from its original full set of 12 orders, using the combination of fuzzy ARTMAP and genetic algorithm. To further reduce the computation time, the combination of q-recursive method and Zernike polynomials’ inherent symmetry property is utilized. Hence, the processing time of the subset of ZM is significantly reduced by almost 67% while maintaining the classification accuracy as compared to computing the original full set of ZMs.

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

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.

Zernike Moments (ZMs) are used in many image processing applications, due to their resistance against various signal processing and geometric attacks. Digital image watermarking is one of those application areas, where ZMs are widely used to insert and extract the watermark bits for digital media authentication. In all the existing ZM based watermarking techniques, magnitude of moments is used to insert and extract the watermark. In this paper, the authors’ have proposed a semi blind watermarking technique in which phase of ZMs is used for watermark insertion and extraction. Due to the use of phase of ZMs, 100% detection ratio is achieved against any geometric and other signal processing attacks. To make the proposed technique fast, q-recursive method is used to compute the Zernike polynomials. The use of q-recursive method has also increased the transparency of watermark due to its better reconstruction ability as compared to traditional moment computation method. Through detailed experimentation, it has been confirmed that the proposed watermarking technique is fast, has more imperceptibility, less Bit Error Rate (BER) and more capacity as compared to traditional ZMs magnitude based watermarking technique.