Accurate image rotation using hermite expansions

Local geometric deformations in the DHT domain with applications

Local rotation, translation and scaling of the image domain represent a basic toolkit in adaptive image processing such as image registration, template matching, local invariant feature detection, super-resolution imaging, among others. In this article, it is shown how the local rotation, scaling and translations can be performed in the discrete Hermite transform (DHT) domain. As the DHT satisfies the generalized steerability property, basic geometric operations are expressed as linear mappings in the DHT domain and hence can facilitate the solution of many image processing problems. The local rotation and scaling were previously shown in the continuous domain using the Hermite Transform, the former is used here as a good approximation for discrete images, whereas the latter is extended to a discrete domain. In addition, the local translation operation is fully developed in the discrete domain. The application of these three operations is illustrated with three exemplar applications including 1) mathematical morphology, 2) template matching and 3) depth from defocus. The simple yet effective methods presented in the paper indicate that local image decompositions satisfying the steerability property, such as the DHT, are desirable for solving a number of interesting image processing problems.

Double Line Image Rotation

This paper proposes a fast algorithm for rotating images while preserving their quality. The new approach rotates images based on vertical or horizontal lines in the original image and their rotated equation in the target image. The proposed method is a one-pass method that determines a based-line equation in the target image and extracts all corresponding pixels on the base-line. Floating-point multiplications are performed to calculate the base-line in the target image, and other line coordinates are calculated using integer addition or subtraction and logical justifications from the base-line pixel coordinates in the target image. To avoid a heterogeneous distance between rotated pixels in the target image, each line rotates to two adjacent lines. The proposed method yields good performance in terms of speed and quality according to the results of an analysis of the computation speed and accuracy.

HermiteFit: fast-fitting atomic structures into a low-resolution density map using three-dimensional orthogonal Hermite functions

HermiteFit, a novel algorithm for fitting a protein structure into a low-resolution electron-density map, is presented. The algorithm accelerates the rotation of the Fourier image of the electron density by using three-dimensional orthogonal Hermite functions. As part of the new method, an algorithm for the rotation of the density in the Hermite basis and an algorithm for the conversion of the expansion coefficients into the Fourier basis are presented. HermiteFit was implemented using the cross-correlation or the Laplacian-filtered cross-correlation as the fitting criterion. It is demonstrated that in the Hermite basis the Laplacian filter has a particularly simple form. To assess the quality of density encoding in the Hermite basis, an analytical way of computing the crystallographic R factor is presented. Finally, the algorithm is validated using two examples and its efficiency is compared with two widely used fitting methods, ADP_EM and colores from the Situs package. HermiteFit will be made available at http://nano-d.inrialpes.fr/software/HermiteFit or upon request from the authors.

STEERABILITY OF HERMITE KERNEL

Steerability is a useful and important property of "kernel" functions. It enables certain complicated operations involving orientation manipulation on images to be executed with high efficiency. Thus, we focus our attention on the steerability of Hermite polynomials and their versions modulated by the Gaussian function with different powers, defined as the Hermite kernel. Certain special cases of such kernel, Hermite polynomials, Hermite functions and Gaussian derivatives are discussed in detail. Correspondingly, these cases demonstrate that the Hermite kernel is a powerful and effective tool for image processing. Furthermore, the steerability of the Hermite kernel is proved with the help of a property of Hermite polynomials revealing the rule concerning the product of two Hermite polynomials after coordination rotation. Consequently, any order of the Hermite kernel inherits steerability. Moreover, a couple sets of an explicit interpolation function and basis function can be directly obtained. We provide some examples to verify steerability of the Hermite kernel. Experimental results show the effectiveness of steerability and its potential applications in the fields of image processing and computer vision.

On the inter-conversion between Hermite and Laguerre local image expansions

The nice relationship existing among the Hermite-Gauss and the Laguerre-Gauss image expansions, whose basis functions span the same signal space, is investigated. As a result, a novel efficient method for Cartesian to polar coordinate inter-conversion, especially suited for dedicated hardware realization, is proposed. Applications to local image rotation, based on the simple steerability of the Laguerre-Gauss expansion, and to local image analysis, based on Gaussian derivatives, are considered in detail. A possible fixed-point realization of the proposed inter-conversion scheme is discussed.

Deblurring of Class-Averaged Images in Single-Particle Electron Microscopy

This paper proposes a method for deblurring of class-averaged images in single-particle electron microscopy (EM). Since EM images of biological samples are very noisy, the images which are nominally identical projection images are often grouped, aligned and averaged in order to cancel or reduce the background noise. However, the noise in the individual EM images generates errors in the alignment process, which creates an inherent limit on the accuracy of the resulting class averages. This inaccurate class average due to the alignment errors can be viewed as the result of a convolution of an underlying clear image with a blurring function. In this work, we develop a deconvolution method that gives an estimate for the underlying clear image from a blurred class-averaged image using precomputed statistics of misalignment. Since this convolution is over the group of rigid body motions of the plane, SE(2), we use the Fourier transform for SE(2) in order to convert the convolution into a matrix multiplication in the corresponding Fourier space. For practical implementation we use a Hermite-function-based image modeling technique, because Hermite expansions enable lossless Cartesian-polar coordinate conversion using the Laguerre-Fourier expansions, and Hermite expansion and Laguerre-Fourier expansion retain their structures under the Fourier transform. Based on these mathematical properties, we can obtain the deconvolution of the blurred class average using simple matrix multiplication. Tests of the proposed deconvolution method using synthetic and experimental EM images confirm the performance of our method.