Moment Functions in Image Analysis — Theory and Applications

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.

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.

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.

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.

Explicit invariance of Cartesian Zernike moments

The comparison of 2-D maps is not trivial, the main difficulties being the high complexity of the sample and the large experimental variability characterizing 2-D gel electrophoresis. The comparison of maps from control and treated samples is usually performed by specific software, providing the so-called spot volume dataset where each spot of a specific map is matched to its analogous in other maps, and they are described by their optical density, which is supposed to be related to the underlying protein amount. Here, a different approach is presented, based on the direct comparison of 2-D map images: each map is decomposed in terms of moment functions, successively applying the multivariate tools usually adopted in image analysis problems. The moments calculated are then treated with multivariate classification techniques. Here, two types of moment functions are presented (Legendre and Zernike moments), while linear discriminant analysis and partial least squares discriminant analysis are exploited as classification tools to provide the classification of the samples. The procedure is applied to a sample dataset to prove its effectiveness.

A method for generating and utilizing structure from motion (SfM) uncertainty estimates within image-based pose estimation is presented. The method is applied to a class of problems in which SfM algorithms are utilized to form a geo-registered reference model of a particular ground area using imagery gathered during flight by a small unmanned aircraft. The model is then used to form camera pose estimates in near real-time from imagery gathered later. The resulting pose estimates can be utilized by any of the other onboard systems (e.g. as a replacement for GPS data) or downstream exploitation systems, e.g., image-based object trackers. However, many of the consumers of pose estimates require an assessment of the pose accuracy. The method for generating the accuracy assessment is presented. First, the uncertainty in the reference model is estimated. Bundle Adjustment (BA) is utilized for model generation. While the high-level approach for generating a covariance matrix of the BA parameters is straightforward, typical computing hardware is not able to support the required operations due to the scale of the optimization problem within BA. Therefore, a series of sparse matrix operations is utilized to form an exact covariance matrix for only the parameters that are needed at a particular moment. Once the uncertainty in the model has been determined, it is used to augment Perspective-n-Point pose estimation algorithms to improve the pose accuracy and to estimate the resulting pose uncertainty. The implementation of the described method is presented along with results including results gathered from flight test data.

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.

Geometric moment invariants to spatial transform and N-fold symmetric blur

Simple and fast computation of moments

Rotation and translation invariants of Gaussian–Hermite moments

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.

Orthogonal invariant Lagrange-Fourier moments for image recognition

with the remarkable growth in image processing, the requirements for dealing out with blurred images is difficulty in a variety of image processing applications. In this paper presents the restoration of blurred images which gets degraded due to diverse atmospheric and environmental conditions, Blur is a key determinant in the sensitivity of image quality, so it is essential to restore the original image. The research outcomes exhibit the major identified bottleneck for restoration is to deal with the blurred images and also a set of attempts have been executed in image restoration using multiple moment algorithms. However the precise results are not been proposed and demonstrated in the comparable researches. Also detail understanding for applications of moment algorithms for image restoration and demonstrating most suitable moment method is current requirements for research. Hence in this work we employ most accepted moment algorithms to exhibit the effect of moments for image restoration and the performance of the moment algorithms such as the Hu, Zernike and Legendre moments is evaluated on image with different blurring lengths. Moreover the effect of moment algorithms is also demonstrated in order to find the optimal setting of orders for image restoration. The final outcome of this work is a stable version of MATLAB based application to visually demonstrate the performance difference of Hu, Zernike and Legendre moments. The relative performance of the application is also been demonstrated with the help of multiple image datasets of biometric identifier such as fingerprint, hand palm and human face.

To improve the accuracy of implant placement in Total Hip Replacement (THR) surgeries, this paper proposes a computer-aided visual system for THR which is composed of a customized acetabular cup, a multi-sensor femoral head trial and a computer for data processing and display. The customized trial is of the same size as the real prosthesis. An image sensor, a gyroscope and an e-compass (including an accelerometer and a magnetometer) are adopted in the femoral head trial. Reference patterns are designed and printed on the internal surface of the cup, whose images are taken by the image sensor for estimation of relative pose and position between the femoral head trial and the acetabulum cup. Two methods of pose estimation are adopted in this system: one based on images and the other based on motion data from gyroscope and e-compass. The efficient perspective-n-point (EPNP) algorithm is used in the image-based pose estimation and achieves a rotation relative error of less than 8% and a translation relative error of less than 10%. The complementary algorithm is adopted in the motion-based pose estimation to smooth the results. Experimental results verified the proposed system.