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Concept# Spherical harmonics

Summary

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
Spherical harmonics originate from

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PurposePatLoc (Parallel Imaging Technique using Localized Gradients) accelerates imaging and introduces a resolution variation across the field-of-view. Higher-dimensional encoding employs more spatial encoding magnetic fields (SEMs) than the corresponding image dimensionality requires, e.g. by applying two quadratic and two linear spatial encoding magnetic fields to reconstruct a 2D image. Images acquired with higher-dimensional single-shot trajectories can exhibit strong artifacts and geometric distortions. In this work, the source of these artifacts is analyzed and a reliable correction strategy is derived. MethodsA dynamic field camera was built for encoding field calibration. Concomitant fields of linear and nonlinear spatial encoding magnetic fields were analyzed. A combined basis consisting of spherical harmonics and concomitant terms was proposed and used for encoding field calibration and image reconstruction. ResultsA good agreement between the analytical solution for the concomitant fields and the magnetic field simulations of the custom-built PatLoc SEM coil was observed. Substantial image quality improvements were obtained using a dynamic field camera for encoding field calibration combined with the proposed combined basis. ConclusionThe importance of trajectory calibration for single-shot higher-dimensional encoding is demonstrated using the combined basis including spherical harmonics and concomitant terms, which treats the concomitant fields as an integral part of the encoding. Magn Reson Med 73:1340-1357, 2015. (c) 2014 Wiley Periodicals, Inc.

Face rendering is a really important topic in Computer Graphics because a lot of virtual simulations or video games contain virtual humans. In order to obtain a realistic face, we need to take care of skin rendering. Nowadays, we can use modern 3D scanning technology to obtain very detailed meshes and textures for the face but the main difficulty with skin rendering is that we need to model subsurface scattering effects. In 2007, d'Eon, Eugene, David Luebke, and Eric Enderton published the article "Efficient Rendering of Human Skin" that describe an algorithm for rendering realistic skin in real-time. The goal of my work was to implement their algorithm to simulate subsurface scattering in skin. I also implemented diffuse environment lighting with occlusions using spherical harmonics.

2011Omnidirectional images are the spherical visual signals that provide a wide, 360◦, view of a scene from a specific position. Such images are becoming increasingly popular in fields like virtual reality and robotics. Compared to conventional 2D images, the storage and badwidth requirements of omnidirectional signals are much higher, due to the specific nature of them. Thus, there is a need for image compression schemes to reduce the dedicated storage space of omnidirectional images. Image compression algorithms can be broadly classified into two groups: lossless and lossy. Lossless schemes are able to reconstruct the exact original data but they cannot reduce the size beyond a specific criteria. Lossy methods are generally better solutions if they do not add a high visual distortion to the reconstructed image, as long as they provide a decent compression rate. If a planar, lossy image compression scheme is applied on omnidirectional images, some problems show up. It is possible to apply a planar compression scheme on a projected version of a 360◦image; however, in these projection schemes (such as equirectangular projection) the sampling rate is different in the poles and the center. Consequently, the filters of the planar compression schemes that do not consider this difference ends in suboptimal result and distortions in the reconstructed images. Recently, with the success of deep neural networks in many image processing tasks, researchers began to use them for the image compression as well. In this study, we propose a deep learning-based method for the compression of omnidirectional images by combining some state of the art approaches from the deep learning-based image compression schemes and some special convolutional layers that take into account the geometry of the omnidirectional image. In comparison to the available methods, it is the first method that can be applied directly on the equirectangularly projected version of omnidirectional images and considers the geometry in the scheme and the layers themselves. To propose this method, different geometry-aware convolutional layers have been tried. We exploited various methods of downsampling and upsampling, such as spherical pooling layers, strided or transposed convolutions, bilinear interpolation, and pixel shuffle. In the end, a method is proposed that benefits from specific spherical convolutional layers which contain sampling methods considering the geometry of omnidirectional images. The sampling positions differ in the different heights of the image based on the nature of the projected omnidirectional image. Additionally, as it benefits from an iterative training method that calculates the residual between the output and input and feeds it again to the network as input of the next iteration, it can provide different compression rates with just one pass of training. Finally, it benefits from a novel method of patching that is well-aligned with the spherical convolution layers and helps the method to run efficiently without a need for a high computational power. The model was compared with a similar architecture without spherical convolutions and spherical patching and showed some improvements. The architecture has been optimized and improved and it has the potential to compete with popular image compression schemes such as JPEG especially in terms of reconstructing the colors.

2020