Practical fast computation of Zernike moments

Simple and fast computation of moments

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.

Pseudo-Zernike moments have better feature representation capability, and are more robust to image noise than those of the conventional Zernike moments. However, due to the computation complexity of pseudo-Zernike polynomials, pseudo-Zernike moments are yet to be extensively used as feature descriptors as compared to Zernike moments. In this paper, we propose two new algorithms, namely coefficient method and p-recursive method, to accelerate the computation of pseudo-Zernike moments. Coefficient method calculates polynomial coefficients recursively. It eliminates the need of using factorial functions. Individual order or index of pseudo-Zernike moments can be derived independently, which is useful if selected orders or indices of moments are needed as pattern features. p-recursive method uses a combination of lower order polynomials to derive higher order polynomials with the same index q. Fast computation is achieved because it eliminates the requirements of calculating polynomial coefficients, Bpqk, and power of radius, rk, in each polynomial. The performance of the proposed algorithms on moment computation and image reconstruction, as compared to those of the present methods, are experimentally verified using a set of binary and grayscale images.

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

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.

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.

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.

The use of 3D discrete orthogonal invariant moments as descriptors of images constitutes one of the hot topics in the field of 3D image analysis, especially in the recognition and classification of deformed objects, due to their high numerical precision and low computing complexity. The TSR (translation, scale and rotation) invariant functions of discrete orthogonal moments can be obtained by expressing them as a linear combination of the corresponding invariant of geometric moments. This classical method is very expensive in time due to the long time allocated to compute the polynomial coefficients when complex or large 3D images are used. This main drawback eliminates the use of these moments in the recognition and classification fields of 3D objects. In this article, we propose a fast and efficient method for calculating 3D discrete orthogonal invariant moments of Hahn, Tchebichef, Krawtchouk, Meixner and Charlier of 3D image. This proposed method is based on the extraction of the 3D discrete orthogonal invariant moments from the 3D geometric invariant moments of the entire 3D image, using the 3D image cuboid representation (ICR) strategy, where each cuboid is treated independently. The proposed method has the advantage of accelerating the process of extracting the 3D discrete invariant moments compared with the classical method. The performance of the proposed method is evaluated in terms of the calculation time and classification accuracy using a set of 3D images. Instead of some examples of using the discrete orthogonal invariant moments of Hahn, Tchebichef, Krawtchouk, Meixner and Charlier as pattern features for pattern classification are also provided.

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.

The advances in biomedical imaging equipment have produced a massive amount of medical images that are generated by the different modalities. Consequently, a huge volume of data has been produced and caused a complex and time-consuming retrieving process of the relevant cases. To resolve this issue, the Content-Based Biomedical Image Retrieval (CBMIR) system is applied to retrieve the related images from the earlier patients’ databases. However, the previous handcrafted features methods that applied the CBMIR model have shown poor performance in many multimodal databases. In this paper, we focus on designing CBMIR technique using Deep Learning (DL) models. We present a new Multimodal Biomedical Image Retrieval and Classification (M-BMIRC) technique for retrieving and classifying the biomedical images from huge databases. The proposed M-BMIRC model involves three dissimilar processes as following: feature extraction, similarity measurement, and classification. It uses an ensemble of handcrafted features from Zernike Moments (ZM) and deep features from Deep Convolutional Neural Networks (DCNN) for feature extraction process. Additionally, the Hausdorff Distance based similarity measure is employed to identify the resemblance between the queried image and the images that exist in the database. Moreover, the classification process gets executed on the retrieval images using the Probabilistic Neural Network (PNN) model, which allocates the class labels of the tested images. Finally, the experimental studies are conducted using two benchmark medical datasets and the results ensure the superior performance of the proposed model in terms of different measures include Average Precision Rate (APR), Average Recall Rate (ARR), F-score, accuracy, and Computation Time (CT).

Zernike moments, as a representative orthogonal moment, have been widely applied in the fields of image processing and pattern recognition. The calculations are time-consuming due to the complexity of definition. Based on the GPU octant symmetry algorithm in our previous work, this paper presents a novel algorithm to increase the resource utilization by the combined kernel. Also, it optimizes radial polynomials of Zernike moments to reduce amount of calculations. The experimental results demonstrated that the proposed algorithm achieved overall computational performance improvement for any sized images. Moreover, there is no compromise in terms of precision compared to the typical accurate algorithm.

This work presents a new approach and an algorithm for binary image representation, which is applied for the fast and efficient computation of moments on binary images. This binary image representation scheme is called image block representation, since it represents the image as a set of nonoverlapping rectangular areas. The main purpose of the image block representation process is to provide an efficient binary image representation rather than the compression of the image. The block represented binary image is well suited for fast implementation of various processing and analysis algorithms in a digital computing machine. The two-dimensional (2-D) statistical moments of the image may be used for image processing and analysis applications. A number of powerful shape analysis methods based on statistical moments have been presented, but they suffer from the drawback of high computational cost. The real-time computation of moments in block represented images is achieved by exploiting the rectangular structure of the blocks.

Moments are important features used in pattern recognition, image analysis and computer vision. Geometric moments are calculated using window functions with great discontinuities at window boundary and the bases are not orthogonal. In order to better characterize noisy images, one should use the orthogonal moments with a smoothing window function. In the present paper, by use of the well-known Gaussian functions as smoothing widow function, we first introduce Gaussian-Hermite moments which are shown to be orthogonal smoothed moments. Then we present the recursive calculation of Gaussian-Hermite moments. The recursive calculation not only gives an efficient scheme for Gaussian-Hermite moment calculation, but also reveals the essential relation between orthogonal Gaussian- Hermite moments and Gaussian-filtered signal derivatives. From this relation, we see that the Gaussian-Hermite moments are good local features for noisy signal. Moreover, because Gaussian derivatives satisfy the conditions for mother wavelets, Gaussian-Hermite moments defined from Gaussian functions of different (sigma) correspond to wavelet development of the input signal. In fact, the n-order Gaussian-Hermite moment is a linear combination of different Gaussian-derivative wavelets of the input signal. So the use of Gaussian-Hermite moments gives also an efficient approach to representing the input signal in an orthogonal functional space from wavelet analysis. All the results above obtained for 1-D signals are generalized to 2-D image analysis.

On the computational aspects of affine moment invariants for gray-scale images