Set Of Affine Transformations Research Articles

Fourier reconstruction for diffraction tomography of an object rotated into arbitrary orientations

In this paper, we study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a microscopic rigid particle rotating in a trap created, for instance, by acoustic or optical forces. Under the influence of the inhomogeneous forces the particle carries out a time-dependent smooth, but irregular motion described by a set of affine transformations. The rotation of the particle enables one to record optical images from a wide range of angles, which largely eliminates the ‘missing cone problem’ in optics. This advantage, however, comes at the price that the rotation axis in this scenario is not fixed, but continuously undergoes some variations, and that the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. In the present work, we assume that the time-dependent motion parameters are known, and that the particle’s scattering potential is compatible with making the first order Born or Rytov approximation. We prove a Fourier diffraction theorem and derive novel backpropagation formulae for the reconstruction of the scattering potential, which depends on the refractive index distribution inside the object, taking its complicated motion into account. This provides the basis for solving the ODT problem with an efficient non-uniform discrete Fourier transform.

New Robust Regularized Shrinkage Regression for High-Dimensional Image Recovery and Alignment via Affine Transformation and Tikhonov Regularization

In this work, a new robust regularized shrinkage regression method is proposed to recover and align high-dimensional images via affine transformation and Tikhonov regularization. To be more resilient with occlusions and illuminations, outliers, and heavy sparse noises, the new proposed approach incorporates novel ideas affine transformations and Tikhonov regularization into high-dimensional images. The highly corrupted, distorted, or misaligned images can be adjusted through the use of affine transformations and Tikhonov regularization term to ensure a trustful image decomposition. These novel ideas are very essential, especially in pruning out the potential impacts of annoying effects in high-dimensional images. Then, finding optimal variables through a set of affine transformations and Tikhonov regularization term is first casted as mathematical and statistical convex optimization programming techniques. Afterward, a fast alternating direction method for multipliers (ADMM) algorithm is applied, and the new equations are established to update the parameters involved and the affine transformations iteratively in the form of the round-robin manner. Moreover, the convergence of these new updating equations is scrutinized as well, and the proposed method has less time computation as compared to the state-of-the-art works. Conducted simulations have shown that the new robust method surpasses to the baselines for image alignment and recovery relying on some public datasets.

An efficient locally affine framework for the smooth registration of anatomical structures

Intra-subject and inter-subject nonlinear registration based on dense transformations requires the setting of many parameters, mainly for regularization. This task is a major issue, as the global quality of the registration will depend on it. Setting these parameters is, however, very hard, and they may have to be tuned for each patient when processing data acquired by different centers or using different protocols. Thus, we present in this article a method to introduce more coherence in the registration by using fewer degrees of freedom than with a dense registration. This is done by registering the images only on user-defined areas, using a set of affine transformations, which are optimized together in a very efficient manner. Our framework also ensures a smooth and coherent transformation thanks to a new regularization of the affine components. Finally, we ensure an invertible transformation thanks to the Log-Euclidean polyaffine framework. This allows us to get a more robust and very efficient registration method, while obtaining good results as explained below. We performed a qualitative and quantitative evaluation of the obtained results on two applications: first on atlas-based brain segmentation, comparing our results with a dense registration algorithm. Then the second application for which our framework is particularly well suited concerns bone registration in the lower-abdomen area. We obtain in this case a better positioning of the femoral heads than with a dense registration. For both applications, we show a significant improvement in computation time, which is crucial for clinical applications.

Querying time series data based on similarity

We study similarity queries for time series data where similarity is defined, in a fairly general way, in terms of a distance function and a set of affine transformations on the Fourier series representation of a sequence. We identify a safe set of transformations supporting a wide variety of comparisons and show that this set is rich enough to formulate operations such as moving average and time scaling. We also show that queries expressed using safe transformations can efficiently be computed without prior knowledge of the transformations. We present a query processing algorithm that uses the underlying multidimensional index built over the data set to efficiently answer similarity queries. Our experiments show that the performance of this algorithm is competitive to that of processing ordinary (exact match) queries using the index, and much faster than sequential scanning. We propose a generalization of this algorithm for simultaneously handling multiple transformations at a time, and give experimental results on the performance of the generalized algorithm.

Applications of Partitioned Iterated Function Systems in Image and Video Compression

Iterated function systems (IFS) have been used to compress image data. Because of difficulty in finding IFS in natural images, a technique based on partitioned IFS (PIFS) has been proposed for image compression. In this technique, an image to be compressed is divided into nonoverlapping blocks. For each block an affine transformation is found in the image. This set of affine transformations (called PIFS) corresponds to a unique image. In the simplest case, images are partitioned into fixed size blocks. In this paper, we investigate image and video compression techniques using variable block sizes based on the quadtree partition. One property of images generated using PIFS is scalability: they have fine detail in any scale. We exploit this property to reduce required compression time and improve compression performance. There are large amounts of temporal redundancy between fames of a video sequence. We describe a method to remove temporal redundancies effectively using a quadtree partitioning technique. We have implemented the above schemes to compress image and video sequences and will report our experimental results.

Extraction of symmetry properties using correlation with rotated and reflected images

AbstractThe symmetrical properties of an image or a figure are important in visual psychology and computer vision.This paper proposes a new method to extract the properties of rotational symmetry and reflectional symmetry from planar two‐dimensional images. In the detection of the rotational symmetry, first, the original image for a quantized range of angles of rotation is rotated. The rotated results are convolved with the image using directional features. Then a set of affine transformations is applied to the parameter space to obtain the correlation results using each pixel as the center of rotation.The periodicity of the correlation along the rotation angle for each point is evaluated. Finally, the local maximum correlation is determined that indicates the center and the number of folds of the rotational symmetry. In the detection of the reflectional symmetry, the image for the lines in the possible quantized directions is reflected. Then the correlation between the resulting images and the original image is calculated.The transformation that maximizes the correlation directly indicates the direction of the axis of the reflectional symmetry and the translation amount of the glide symmetry. This method does not require any segmentation nor knowledge of the centroid position.The method in this paper evaluates the transformation that matches the images even though the match is a partial one; thus, it can extract the symmetrical property that exists only partially.Although a large amount of computation and storage capacity is required, the method is fundamental, versatile, and suited to the parallel computation.

Affine and general linear equivalences of boolean functions

An algorithm is presented which determines, given two Boolean functions, the set of affine transformations and the set of general linear transformations under which the functions are equivalent. A method for detecting linear functions is developed for use in the algorithm.