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Concept# Orthographic projection

Summary

Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.
The term orthographic sometimes means a technique in multiview projection in which principal axes or the planes of the subject are also parallel with the projection plane to create the primary views. If the principal planes or axes of an object in an orthographic projection are not parallel with the projection plane, the depiction is called axonometric or an auxiliary views. (Axonometric projection is synonymous with parallel projection.) Sub-types of primary views include plans, elevations, and sections; sub-types of auxiliary views include isometric, dimetric, and trimetric projections.
A lens that provides an orthographic projection is an object-space telecentric lens.
A simple orthographic projection onto the plane z = 0 can be defined by the following matrix:
For each point v = (vx, vy, vz), the transformed point Pv would be
Often, it is more useful to use homogeneous coordinates. The transformation above can be represented for homogeneous coordinates as
For each homogeneous vector v = (vx, vy, vz, 1), the transformed vector Pv would be
In computer graphics, one of the most common matrices used for orthographic projection can be defined by a 6-tuple, (left, right, bottom, top, near, far), which defines the clipping planes. These planes form a box with the minimum corner at (left, bottom, -near) and the maximum corner at (right, top, -far).
The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1).

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Orthographic projection

Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.

3D projection

A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane. 3D projections use the primary qualities of an object's basic shape to create a map of points, that are then connected to one another to create a visual element.

Computer graphics

Computer graphics deals with generating s and art with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. A great deal of specialized hardware and software has been developed, with the displays of most devices being driven by computer graphics hardware. It is a vast and recently developed area of computer science. The phrase was coined in 1960 by computer graphics researchers Verne Hudson and William Fetter of Boeing.

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Omnidirectional 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.

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