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Perspective Picture from Visual Sphere: a New Approach to Image Rasterization

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Perspective Picture from Visual Sphere: a New Approach to Image Rasterization Perspective picture from Visual Sphere: a new approach to image rasterization Jakub Maximilian Fober∗ January 7, 2020 S 1 Introduction → A This paper presents new perspective model. One based on a Gaussian Sphere, able to reproduce G image of any shape, perspective geometry and an- gle of view (AOV). A model capable of combin- ing pictures from sources previously incompati- ble, like fish-eye camera view and wide-angle lens h picture. Based on this model, new rasterization → B method is presented, one able to render real-time → C curvilinear image directly from third-dimension. This work is divided into nine sections, grad- ually unfolding the topic, from philosophical Figure 1: Projection of ABC triangle onto image standpoint, to technical specifics. In following grid G of a visual unit sphere S. Projected edge section, the history of perspective picture is in- CA is formed by the great circle h. troduced, establishing grounds for later discus- sion. Sections three and four overview percep- tion of perspective picture, its geometry and sen- Abstract sation of motion. Following sections five and six present Universal Perspective model. Fifth There is great demand for perspective projection section relates to mathematics of perspective; model able to produce computer-generated (CG) present equations of transformation and projec- image up to 360° of view, with lens-distortions, tion of 2D/3D data and lens distortions. Sixth directly from three-dimensional space, to pixel section presents rasterization process for 3D sur- data. Currently there is no practical direct faces. Section number seven refers to generat- method for rasterization of real-time graphics in ing barycentric coordinates of a triangle in the curvilinear perspective. Every real-time perspec- Universal Perspective model. Section eight in- tive imagery incorporates Pinhole Camera model corporates Universal Perspective model in mea- as a base, some with additional layers of distor- surement and simulation of real optical systems, tion on top. Also to note, knowledge about rela- with variable no-parallax point. Section nine fol- tionship between motion and perspective has not lowed by references, conclude all information and been properly formulated, leaving a void in that present direction for further studies. All support field of image science. graphical material is presented on last five pages. This paper aims at solving those issues. Study involves exploring history of perspective picture, redefining abstract theorem of image (as recorded in common-knowledge), establishing rules of per- spective image and presenting new, universal model for perspective projection and rasteriza- tion in CG graphics. Keywords: perspective, non-linear projec- Figure 2: Rasterized polygon with texture (Paolo tions, spherical perspective, graphics hardware, th mathematics of art Uccello, 15 century) from a perspective map, where Ωd = 270°, k = 0:32, l = 62%, s = 86%. ∗ [email protected] & http://maxfober.space/ 1 2 Previous work and history of mixed spherical and cylindrical projection in of the topic anamorphic lens, with perspective preservation. Definition. Conservation of perspective - lines Current image abstract theorem was established th converging at the optical-axis vanishing-point re- in 15 century, in a book De Pictura, by main straight (also see perspective picture defini- L. B. Alberti. Based on invention of F. tion on the next page). Brunelleschi, Alberti defined geometrical and theoretical rules for designing perspective CG image technology did not follow film indus- projections.2 These rules are currently used in try progress in that field. Still based on Alberti CG polygon-based graphics. theorem, computer graphics became incompati- Major theoretical statement that laid founda- ble with film, generating great costs, when two 20 tion for image projection technologies and current had to be joined together. Which part took lens understanding of image nature can be found in aberration rotoscopy, where geometry correction Alberti abstract definition of image. He would has to be performed manually at each frame. describe a painting to be like a window in a wall.1 Currently in computer-games industry, CG im- agery is practically unable to produce realistic, “First of all, on the surface on which I curvilinear simulation of visual space (VS), or am going to paint, I draw a rectangle of even simulate anamorphic lens geometry, due to whatever size I want, which I regard as limits of linear perspective and resource costs an open window through which the sub- of overcoming those. Some hybrid solutions for 1 ject to be painted is seen.” real-time graphics where proposed,11,12 that com- bine rasterization with ray-tracing, or tessella- But in times of its discovery, as for now, linear tion. Such approach allows for a semi-practical perspective introduced itself with several issues. and limited production of real-time pictures in a When there is a need for a wide-angle view, one non-linear perspective. close to the human visual field, geometrical dis- tortions appear to dominate visual aspect of the picture. Those issues were noticed by Renais- 3 On visual space geometry sance artists, like L. Da Vinci. He put to test the Alberti theorem and produced paintings of and image perception accurate-perspective.6 In his Treatise on Paint- Perspective picture visible inside the visual space ing, Da Vinci notes that picture conforms to the gives some sense of immersion (e.g. picture, film, idea of a window only when viewed from one spe- computer game) even without visual illusion.8 It cific point in space.7 Stating that seen otherwise, is perceived as a visual symbol of an abstract objects appear distorted, especially in the periph- point of view. Picture immersion does not break, ery. Picture then, viewed from a different point as long as object’s appearance do not exhibit too ceases to be like a window in a wall and becomes much deformation. a visual symbol of an abstract point of view.a Representing point of view invokes separation Some 18th century late Baroque and Neo- from the surrounding. To enhance this effect, en- classical artists, when encountered those is- vironment stimuli is being reduced, like in case sues, introduced derivative projections. Like of movie theater. To uphold the immersion lights G. P. Pannini with later re-discovered Panini are turned off and silence is expected. Or in case Projection,17 or R. Barker, who established of horror-game session played at night, to sepa- the term Panorama.25 It was a new type of rate from safe-space of home. perspective, in a form of cylindrical projection, where abstract window frame became horizon- Remark. On the opposite side, picture which is tally curved, reducing deformation artifacts in a an integral part of the surrounding, can be cate- 23 wide, panoramic depictions of architecture. gorized under the Trompe l’œil technique. Since most of the time picture presents point Invention of motion picture followed by the of view (e.g. film, video game, visualiza- rise of film industry, resulted in demand for a tion), it’s wise to consider subject’s properties new image geometry. Previously still, now pic- of vision when designing picture’s perspective. tures had to be pleasing to the eye, in motion. But instead of producing mechanical simula- 1950s brought anamorphic cinematography to the tion, perspective should symbolize total sensory 2 wider audience. Lenses like CinemaScope and experience. later Panavision14 became a standard in film Theorem. To create immersive visual symbol of production. Figure 3 on page 11 shows example a visual space, it is necessary to use the curvilin- aEffect also referred to Zeeman Paradox.8 ear perspective instead of linear. 2 Proof. Geometry of human visual space contra- panini, to symbolize binocular vision; two dicts linear perspective principle, as visual field spherical projections combined into extends beyond linear perspective angle of view. one panoramic image.b Linear perspective, based on a tangent of an an- gle, exhibits limit of 179: (9) 8° of view. While anamorphic, as cylindrical projections, like visual field extends horizontally up to 220° for Panini, tend to elongate propor- 13 binocular vision. tions vertically; there is a need for correction. At narrow AOV both types of perspective are suitable for immersive picture. In such case dif- Visual sphere as a whole image ferences are negligible. Sub-figure 4a on page 11 presents these differences in comparison to five Common idea of an image is limited to a finite, major perspective projections. Same differences two-dimensional plane. Which is subjective, due seem exaggerated at higher AOV values (see Sub- to constrains of human visual field and up-front figure 4b). placement of eyes. One can construct a rectan- Corollary. Practical limit for immersive pic- gular frame, which at certain distance from the ture in linear perspective is between 60° and 110° eyes will cover full visual field (VF). In case of some animals (e.g. horse, rabbit), visual space AOV. Wider-angles exhibit deformations known 3,16 as the Leonardo Paradox,8 which are then domi- confines much wider VF. With only few blind nant in the image perception and break picture’s spots, spanning to almost 360°. Such field cannot immersion. be enclosed by a single rectangular frame. Thus image nature is not of a frame. Another model To show wider-angle picture it is necessary to has to be chosen instead. One able to cover full use curvilinear projection. But there is a ten- Ω = 360° is a sphere. dency to see the world not through sight but the understanding. We understand that the wall is Remark. Cylindrical projections cannot cover full 360 flat, therefore we see it that way. Picture pro- ° of view, in all directions. They are a hy- jected into the eye is just a visual symbol of a brid between frame and spherical model.
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