Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.

PatchWarp: Corrections of non-uniform image distortions in two-photon calcium imaging data by patchwork affine transformations

Two-photon microscopy has been widely used to record the activity of populations of individual neurons at high spatial resolution in behaving animals. The ability to perform imaging for an extended period of time allows the investigation of activity changes associated with behavioral states and learning. However, imaging often accompanies shifts of the imaging field, including rapid (~100ms) translation and slow, spatially non-uniform distortion. To combat this issue and obtain a stable time series of the target structures, motion correction algorithms are commonly applied. However, typical motion correction algorithms are limited to full field translation of images and are unable to correct non-uniform distortions. Here, we developed a novel algorithm, PatchWarp, to robustly correct slow image distortion for calcium imaging data. PatchWarp is a two-step algorithm with rigid and non-rigid image registrations. To correct non-uniform image distortions, it splits the imaging field and estimates the best affine transformation matrix for each of the subfields. The distortion-corrected subfields are stitched together like a patchwork to reconstruct the distortion-corrected imaging field. We show that PatchWarp robustly corrects image distortions of calcium imaging data collected from various cortical areas through glass window or GRIN lens with a higher accuracy than existing non-rigid algorithms. Furthermore, it provides a fully automated method of registering images from different imaging sessions for longitudinal neural activity analyses. PatchWarp improves the quality of neural activity analyses and would be useful as a general approach to correct image distortions in a wide range of disciplines.

New Robust Part-Based Model with Affine Transformations for Facial Landmark Localization and Detection in Big Data

In this paper, we developed a new robust part-based model for facial landmark localization and detection via affine transformation. In contrast to the existing works, the new algorithm incorporates affine transformations with the robust regression to tackle the potential effects of outliers and heavy sparse noises, occlusions and illuminations. As such, the distorted or misaligned objects can be rectified by affine transformations and the patterns of occlusions and outliers can be explicitly separated from the true underlying objects in big data. Moreover, the search of the optimal parameters and affine transformations is cast as a constrained optimization programming. To mitigate the computations, a new set of equations is derived to update the parameters involved and the affine transformations iteratively in a round-robin manner. Our way to update the parameters compared to the state of the art of the works is relatively better, as we employ a fast alternating direction method for multiplier (ADMM) algorithm that solves the parameters separately. Simulations show that the proposed method outperforms the state-of-the-art works on facial landmark localization and detection on the COFW, HELEN, and LFPW datasets.

On the Impact of Affine Loop Transformations in Qubit Allocation

Most quantum compiler transformations and qubit allocation techniques to date are either peep-hole focused or rely on sliding windows that depend on a number of external parameters including the topology of the quantum processor. Thus, global optimization criteria are still lacking. In this article, we explore the synergies and impact of affine loop transformations in the context of qubit allocation and mapping. With this goal in mind, we designed and implemented AXL , a domain specific language and source-to-source compiler for quantum circuits that can be directly described with affine relations. We conduct an extensive evaluation spanning circuits from the recently introduced QUEKO benchmark suite, eight quantum circuits taken from the literature, three distinct coupling graphs, four affine transformations (including the Pluto dependence distance minimization and Feautrier’s minimum latency algorithms), four qubit allocators, and two back-end quantum compilers. Our results demonstrate that affine transformations using global optimization criteria can cooperate effectively in several scenarios with quantum qubit mapping algorithms to reduce the circuit depth, size and allocation time.

New Robust PCA for Outliers and Heavy Sparse Noises’ Detection via Affine Transformation, the L ∗ , w and L 2,1 Norms, and Spatial Weight Matrix in High-Dimensional Images: From the Perspective of Signal Processing

In this paper, we propose a novel robust algorithm for image recovery via affine transformations, the weighted nuclear, L ∗ , w , and the L 2,1 norms. The new method considers the spatial weight matrix to account the correlated samples in the data, the L 2,1 norm to tackle the dilemma of extreme values in the high-dimensional images, and the L ∗ , w norm newly added to alleviate the potential effects of outliers and heavy sparse noises, enabling the new approach to be more resilient to outliers and large variations in the high-dimensional images in signal processing. The determination of the parameters is involved, and the affine transformations are cast as a convex optimization problem. To mitigate the computational complexity, alternating iteratively reweighted direction method of multipliers (ADMM) method is utilized to derive a new set of recursive equations to update the optimization variables and the affine transformations iteratively in a round-robin manner. The new algorithm is superior to the state-of-the-art works in terms of accuracy on various public databases.

Space-Time Loop Tiling for Dynamic Programming Codes

We present a new space-time loop tiling approach and demonstrate its application for the generation of parallel tiled code of enhanced locality for three dynamic programming algorithms. The technique envisages that, for each loop nest statement, sub-spaces are first generated so that the intersection of them results in space tiles. Space tiles can be enumerated in lexicographical order or in parallel by using the wave-front technique. Then, within each space tile, time slices are formed, which are enumerated in lexicographical order. Target tiles are represented with multiple time slices within each space tile. We explain the basic idea of space-time loop tiling and then illustrate it by means of an example. Then, we present a formal algorithm and prove its correctness. The algorithm is implemented in the publicly available TRACO compiler. Experimental results demonstrate that parallel codes generated by means of the presented approach outperform closely related manually generated ones or those generated by using affine transformations. The main advantage of code generated by means of the presented approach is its enhanced locality due to splitting each larger space tile into multiple smaller tiles represented with time slices.

Automated combination of optical coherence tomography images and fundus images

We discuss approaches to combining multimodal multidimensional images, namely, three-dimensional optical coherence tomography (OCT) data and two-dimensional color images of the fundus. Registration of these two modalities can help to adjust the position of the obtained OCT images on the retina. Some existing approaches to matching fundus images are based on finding key points that are considered invariant to affine transformations and are common to the two images. However, errors in the identification of such points can lead to registration errors. There are also methods for iterative adjustment of conversion parameters, but they are based on some manual settings. In this paper, we propose a method based on a full or partial search of possible combinations of the OCT image transformation to find the best approximation of the true transformation. The best approximation is determined using a measure of comparison of preprocessed image pixels. Further, the obtained transformations are compared with the available true transformations to assess the quality of the algorithm. The structure of the work includes: pre-processing of OCT and fundus images with the extraction of blood vessels, random search or grid search over possible transformation parameters (shift, rotation and scaling), and evaluation of the quality of the algorithm.

Multimodal Tucker Decomposition for Gated RBM Inference

Gated networks are networks that contain gating connections in which the output of at least two neurons are multiplied. The basic idea of a gated restricted Boltzmann machine (RBM) model is to use the binary hidden units to learn the conditional distribution of one image (the output) given another image (the input). This allows the hidden units of a gated RBM to model the transformations between two successive images. Inference in the model consists in extracting the transformations given a pair of images. However, a fully connected multiplicative network creates cubically many parameters, forming a three-dimensional interaction tensor that requires a lot of memory and computations for inference and training. In this paper, we parameterize the bilinear interactions in the gated RBM through a multimodal tensor-based Tucker decomposition. Tucker decomposition decomposes a tensor into a set of matrices and one (usually smaller) core tensor. The parameterization through Tucker decomposition helps reduce the number of model parameters, reduces the computational costs of the learning process and effectively strengthens the structured feature learning. When trained on affine transformations of still images, we show how a completely unsupervised network learns explicit encodings of image transformations.