Pantomorphic Perspective for Immersive Imagery

Home , Planar projection, Stereographic projection

Jakub Maksymilian Fober [email protected]

Abstract even more critical, as there is no mathematical model for gen- A wide choice of cinematic lenses enables motion-picture erating anamorphic projection in an artistically-convincing creators to adapt image visual-appearance to their creative manner [Yuan and Sasian 2009]. Some attempts were made vision. Such choice does not exist in the realm of real-time at alternative projections for computer graphics, with fixed computer graphics, where only one type of perspective pro- cylindrical or spherical geometry [Baldwin et al. 2014; Sharp- jection is widely used. This work provides a perspective less et al. 2010]. A parametrized perspective model was imaging model that in an artistically convincing manner also proposed as a new standard [Correia and Romão 2007], resembles anamorphic photography lens variety and more. but wasn’t adopted. It included interpolation states in be- It presents an asymmetrical-anamorphic azimuthal projec- tween rectilinear/equidistant and spherical/cylindrical pro- tion map with natural vignetting and realistic chromatic jection. The cylindrical parametrization of this solution was aberration. The mathematical model for this projection has merely an interpolation factor, where intermediate values been chosen such that its parameters reflect the psycho- did not correspond to any common projection type. There- physiological aspects of visual perception. That enables its fore it was not suited for artistic or professional use. The use in artistic and professional environments, where specific notion of sphere as a projective surface, which incorporates aspects of the photographed space are to be presented. cartographic mapping to produce a perspective picture be- came popularized [German et al. 2007; Peñaranda et al. 2015; CCS Concepts: • Computing methodologies → Percep- Williams 2015]. Also, perspective parametrization that tran- tion; Ray tracing; • Human-centered computing; • Ap- sitions according to the content (by the view-tilt angle) has plied computing → Media arts; been developed, as a modification to the computer game Keywords: Curvilinear Perspective, Panini, Pantomorphic, Minecraft [Williams 2017]. But the results of these solutions Cylindrical, Fisheye, Perspective Map were more a gimmick and have not found practical use. The linear perspective projection is still the way-to-go for most digital content. One of the reasons is the GPU fixed-function © 2021 Jakub Maksymilian Fober architecture in regard to rasterization. With the advent of real-time ray-tracing, more exotic projections could become widely adopted. This work is licensed under Creative Commons BY-NC-ND 3.0 license. https://creativecommons.org/licenses/by-nc-nd/3.0/ This paper aims to provide a perspective model with math- For all other uses including commercial, contact the owner/author(s). ematical parametrization that gives artistic-style interaction with image-geometry. It is a parametrization that works similarly to the way film directors choose lenses for each 1 Introduction scene by their aesthetics [Giardina 2016; Neil 2004; Sasaki The perspective in computer real-time graphics hasn’t changed 2017a], but with a greater degree of control. It also provides a since the dawn of CGI. It is based on a concept as old as the psycho-physiological correlation between perspective model fifteenth century Renaissance, a linear projection [Alberti parameters and the perception of depicted space attributes, like distance, size, proportions, shape or speed. That map- arXiv:2102.12682v4 [cs.GR] 19 Jul 2021 1435; Argan and Robb 1946; McArdle 2013]. This situation is similar to the beginnings of photography, where only one ping enables the use of the presented model in a professional type of lens was widely used, an A.D. 1866 Rapid Rectilinear environment, where image geometry is specifically suited [Kingslake 1989] lens. Linear perspective even at the time of for a task [Whittaker 1984]. pantomorphic its advent, 500 years ago, has been criticized for distorting The presented perspective model is named panto- +morphe¯ proportions [Da Vinci 1632]. In a phenomenon known today (G. , all , shape; assuming all shapes). as Leonardo’s paradox [Dixon 1987], in which figures further away but at the periphery appear larger, than those located 1.1 Document naming convention near the optical center. Computer graphics really skipped This document uses the following naming convention: the artistic achievements of the last five centuries in regard to perspective. This includes the cylindrical perspective of • Left-handed coordinate system. Pannini [Sharpless et al. 2010], Barker [Wikipedia, contribu- • Vectors presented natively in column. tors 2019] and anamorphic lenses used in cinematography • Matrix vectors arranged in rows (row-major order) [Giardina 2016; Neil 2004; Sasaki 2017a,b]. The situation is 푀row col. Fober, J.M.

ì2 푣푥 ( ) • Matrix-matrix multiplication by [column]푎 ·[row]푏 =  2 2  " 1 cos 2휑 # 휑ì 푣ì +ì푣 cos2 휑 + 푀 푥 =  푥 푦  = = 2 2 , 푎 푏. 휑  푣ì2  2 휑 cos(2휑) (1d) ì푦  푦  sin 1 − • |푢| 2 2 2 2 A single bar enclosure “ ” represents vector length  푣ì푥 +ì푣푦  or scalar absolute.   • Vectors with an arithmetic sign, or without, are calcu- where 푟 ∈ R>0 is the view-coordinate radius (magnitude). lated component-wise and form another vector. Vector 휃ì ∈ [0, 휋]2 contains incident angles (measured from • Centered dot “·” represents the vector dot product. the optical axis) of two azimuthal projections. Vector 휑ì ∈ 푓 , 푐 • Square brackets with a comma “[ ]” denote interval. [0, 1]2 is the anamorphic interpolation weight, which is lin- • Square brackets with blanks “[푥 푦]” denote vector and ear 휑ì푥 +휑 ì푦 = 1, but has spherical distribution (see Figure 1 matrix. on the facing page). Vector 푘ì ∈ [−1, 1]2 (also [−1, 1]3 in sub- • The power of “−1” signifies the reciprocal of the value. section 2.2 variant) describes two power axes of anamorphic • QED symbol “□” marks the final result or output. projection. The algorithm is evaluated per-pixel of position This naming convention simplifies the transformation pro- 푣ì ∈ R2 in view-space coordinates, centered at the optical axis cess of formulas into code. and normalized at the chosen angle-of-view (horizontal or vertical). The final anamorphic incident angle 휃 ′ is obtained 2 Anamorphic primary-ray map by interpolation, using 휑ì weights. If we assume that projective visual space is spherical [Fleck "휃ì # 휑 1994; McArdle 2013], one can define perspective picture as 휃 ′ 푥 ì푥 = ì · 휑 (2a) an array of rays pointing to the visual sphere surface. This is 휃푦 ì푦 how the algorithm described below will output a projection- 퐺ˆ 푣 휑 map (aka perspective-map). Perspective map is an array of  푥   sin 휃′ ì푥   휃 ′ cos     푟  sin  퐺ˆ푦 =  푣ì푦  =  sin 휑  , □ (2b) three-dimensional rays, which are assigned to each screen-    ′   ′  퐺ˆ   cos휃   cos휃  pixel. Visual sphere as the image model enables a wider  푧      angle of view, beyond 180°-limit of planar projection. Ray- ′ map can be easily converted to cube-푈푉 -map, 푆푇 -map and here 휃 ∈ (0, 휋] is the anamorphic incident angle, measured other screen distortion formats. from the optical axis. This measurement resembles azimuthal Here, the procedural algorithm for primary-ray map (aka projection of the globe (here a visual sphere) [McArdle 2013]. 3 perspective map) uses two input values from the user: dis- The final incident vector 퐺ˆ ∈ [−1, 1] (aka primary-ray) is ′ tortion parameter for two power axes and focal-length or obtained from the anamorphic angle 휃 . Parameters 푟, 푣,ì 휑ì angle-of-view (aka FOV). Two distinct power axes define the are in view-space, while 휃,ì 휃 ′, 휑,퐺ˆ are in visual-sphere space. anamorphic projection type. Each axis is expressed by the Essentially the anamorphic primary-ray map preserves the azimuthal projection factor 푘 [Bettonvil 2005; Fleck 1994; azimuthal angle 휑, while modulating only the picture’s ra- Krause 2019]. Both power axes share the same focal-length dius. 푓 . Evaluation of each power-axis produces spherical an- ì ì gles 휃푥 and 휃푦, which are then combined to anamorphic 2.1 Focal length and angle-of-view ′ azimuthal-projection angle 휃 . Interpolation of 휃 is done To give more control over the picture, a mapping between 휑 휑 through anamorphic weights ì푥 and ì푦. Weights are derived angle-of-view Ω and focal-length 푓 can be established. Here, from spherical angle 휑, of the azimuthal projection. To note, focal-length is derived in a reciprocal from, to optimize usage calculation of 휑 is omitted here, by using view-coordinates in Equation (1). 푣ì-optimization. Ω tan ℎ 푘ì  2 푥 , 푘ì  ì if 푥 > 0 √︃  푘푥 2 2 −1  Ω 푟 = |ì푣| = 푣ì푥 +푣 ì푦 (1a) 푓 ℎ , 푘ì , Ωℎ = 2 if 푥 = 0 (3) Ω 푟  sin ℎ ì arctan 푘ì  2 푘푥  푓 푥 ì  , if 푘ì < 0  , if 푘푥 > 0  ì 푥  푘푥  푘푥 ì  푟 ì 휃푥 = , if 푘푥 = 0 (1b) 푓 Ω ∈ (0, 휏]  arcsin 푟 푘ì where ℎ denotes a horizontal angle of view. Simi-  푓 푥 , 푘ì  if 푥 < 0 larly, vertical Ω can be obtained using 푘ì parameter instead.  푘푥 푣 푦 1 arctan 푟 푘ì The resultant value /푓 ∈ R>0 is the reciprocal focal-length.  푓 푦 , 푘ì  if 푦 > 0  푘푦 휃ì  푟 , 푘ì Remark. Focal-length 푓 value must be same for 휃ì and 휃ì . 푦 = 푓 if 푦 = 0 (1c) 푥 푦  arcsin 푟 푘ì Therefore, only one reference angle Ω can be chosen, either  푓 푦 , 푘ì  if 푦 < 0 horizontal or vertical.  푘푦 Pantomorphic Perspective for Immersive Imagery

r 1 1

0 0 1 r

(a) Graph mapping angle 휑, to anamor- (b) Radial graph, mapping anamorphic (c) Radial graph illustrating anamorphic 2 phic interpolation weights 휑ì푥 and 휑ì푦. interpolation weights 휑ì푥 and 휑ì푦 (as ra- interpolation vector 휑ì ∈ [0, 1] . Here Illustrates circular distribution of 휑ì, in a dius 푟), to angle 휑 in radians. Such that colors red, green, blue represent contri- ì ì ì periodic function. 휑ì푥 +휑 ì푦 = 1. bution of 푘푥 , 푘푦 and 푘푧 respectively.

Figure 1. Illustrating correlation between the anamorphic interpolation weights 휑ì푥 , 휑ì푦 and the spherical angle 휑.

Inverse function to Equation (3), for angle-of-view Ω from distance. For the horizontal power axis, choosing stereo- focal-length 푓 , is obtained as follows graphic projection (which preserves angles and proportions), would enhance cornering. 푘ì  2 arctan 푦  푓 , 푘ì 0  ì if 푦 >  푘푦 2.3 Vignetting mask 푓  2 , 푘ì Ω푣 = 푓 if 푦 = 0 (4)  푘ì Vignetting is an important visual symbol indicating stretch-  2 arcsin 푦  푓 ì  , if 푘푦 < 0 ing of the visual sphere by the projection. Incorporating the  푘ì  푦 vignetting effect enhances space perception. Here, anamorphic vignetting mask Λ′ is obtained similarly This formula can be used to obtain the actual vertical angle- 휃 ′ ì of-view from a horizontally established focal length. Simi- to anamorphic angle . Two separate vignetting masks, Λ푥 and Λì are generated for each power axis and combined to larly horizontal angle Ω can be obtained using the 푘ì param- 푦 ℎ 푥 a single anamorphic vignette Λ′ by interpolation through eter. Diagonal angle Ω can be obtained using interpolation 푑 휑ì-weights. by 휑ì.

2.2 Asymmetrical anamorphism 푘푥 +3 ì  |푘ì |, 1 휃ì 2  ì Λ푥  cos max 푥 2 푥  Parameter 푘 can be further argumented to produce asym- +3 푦 ì =  푘푦  (6a) Λ푦  |푘ì |, 1 휃ì 2  metrical anamorphic projection through the incorporation  cos max 푦 2 푦  푘ì   of a third parameter 푧. In such a case, the bottom and top ì 휑 ′ Λ푥 ì푥 , half of the image can present different azimuthal projection. Λ = ì · 휑 □ (6b) Λ푦 ì푦 ( 푘ì , if 푣ì < 0 푘ì′ = 푧 푦 (5) 푦 푘ì , ′ 푦 otherwise where Λ ∈ [0, 1] is the anamorphic vignetting mask value, interpolated using circular-function-vector 휑ì. 푘ì′ 푘ì therefore 푦 replaces 푦 in equations (1), (4) and (6). The vignette mask of each axis is obtained using two laws Asymmetrical anamorphism can be applied to any power of illumination falloff. Inverse-square law (for 푘 = 1) and axis. The use case for such a perspective would be racing, the cosine law of illumination (for 푘 = −1). The vignette where the bottom-half of the screen contains an image of value for cosine law is simply expressed as cos휃. Inverse- the road. Choosing equidistant projection (which preserves square law value can be expressed as cos2 휃. Therefore the angular speed), would provide an accurate perception of value for projections other than 푘 ∈ {−1, 1} must have a velocity. The top-half of the screen contains the image of vignetting value in-between cos1 휃 ↔ cos2 휃. This has been opponent vehicles. Choosing equisolid projection (which empirically evaluated to a power value linearly-mapped from preserves area) would provide an enhanced perception of 푘 ∈ [−1, 1] ↦→ [1, 2]. Fober, J.M.

Value of 푘 Azimuthal projection type Azimuthal projection type Perception of space

푘푖 = 1 Rectilinear (aka Gnomonic) Rectilinear straightness 푘푖 = 1/2 Stereographic Stereographic shape, angle 푘푖 = 0 Equidistant Equidistant speed, aim 푘푖 = −1/2 Equisolid Equisolid distance, size 푘푖 = −1 Orthographic (azimuthal) Orthographic (azimuthal) — Source: PTGui 11 fisheye factor [Krause 2019]. Source: Empirical study using various competitive video games.

(a) Azimuthal projection type and corresponding 푘 value. (b) Correct perception of space attributes and corresponding azimuthal projection type. Table 1. Tables presenting perspective parameters, corresponding projection type and associated perception attitude.

Picture content type Anamorphic 푘ì values power 2 θ scale Racing simulation 푘ì = [ 1/2 −1/2 0 ] Flying simulation 푘ì = [ −1/2 0 ] Stereopsis (cyclopean) 푘ì = [ 0 −1/2 ] ì 1 First-person aiming 푘 = [ 0 3/4 −1/2 ] ì ì Pan motion 푘푥 ≠ 푘푦 ì ì Roll motion 푘푦 = 푘푥 a ì ì Tilt motion 푘푦 → 푘푥 -1 -0.5 0 0.5 1k Source: Determined empirically using various competitive video games, in accordance to data in Table 1b. aMapping of vertical distortion by a tilt motion introduced first in a Figure 2. Graph illustrating interpolation parameters for Minecraft mod [Williams 2017]. anamorphic vignetting (represented by vertical axis), driven Table 2. 푘ì by parameter 푘 ∈ [−1, 1] (represented by horizontal axis). Recommended values of , for various types of 푘+3 content. The power-graph 2 interpolates between the cosine law of illumination and inverse-square law vignetting. 휃-scale graph scales the vignetting boundary to maximum Ω for given 푘.

4 Anamorphic lens distortion 3 Converting ray-map to ST-map The presented perspective model can be used to mimic the real-world anamorphic lens and its effects. Effects such as dis- Ray/perspective-map can be easily converted to ST-map proportionate lens breathing using focal length-based parametriza- format, given that the maximum view angle Ω does not tion, which are unique to anamorphic photography [Neil exceed or isn’t equal to 180°. 2004; Sasaki 2017b]. Some additional lens-corrections may be added to the initial ray-map, to simulate more complex h i 푤 , lens distortion and lens imperfections.  1 if Ωℎ 푎 푎  ℎ Below an algorithm for anamorphic distortion of view ì푥 ì푦 = h i (7a) ℎ , coordinates is presented, which can be used as an input for  푤 1 if Ω푣  primary-ray map algorithm (Equation 1 on page 2). It is based " # 푓ì cot Ω 퐺ˆ 푎ì 1 on the Brown-Conrady lens-distortion model [Wang et al. 푠 = 2 푥 푥 + , □ (7b) 푓ì 퐺ˆ 푎ì 2008] in a division variant [Fitzgibbon 2001]. It is executed 푡 2퐺ˆ푧 푦 푦 2 on view-coordinates 푣ì, forming alternative vector 푣ì′. where 푎ì ∈ R2 is the square-mapping vector for both the horizontal and vertical angle of view. Values 푤 and ℎ rep- " # 푓ì 푣ì −푐 ì resent picture width and height, respectively. Ω < 휋 is the 푥 = 푥 1 (8a) 푓ì 푣ì −푐 ì angle of view. 푓ì ∈ [0, 1]2 represent the final ST-map vector. 푦 푦 2 퐺ˆ , 3 | {z } ∈ [0 1] is the input primary-ray map vector. cardinal offset a

Pantomorphic Perspective for Immersive Imagery

(a) 푘ì = 0 3/4 − 1/2 , (asymmetrical) first-person. (b) 푘ì = 1/2 − 1/2 0 , (asymmetrical) racing.

Figure 3. Example of various wide-angle (Ω푣 = 100°) anamorphic-azimuthal projections with vignetting in 16:9 aspect-ratio. The checkerboard depicts a cube centered at the view-position, with each face colored according to axis direction. Here, primary colors represent the positive axis and complementary colors its opposite equivalent (same as in the color-wheel), {퐶푦, 푀푔, 푌푙} ← − {푋, 푌, 푍 } + ↦→ {푅,퐺, 퐵}.

2 T 2 휑ì ì2 푓ì lens-transformation. 휑ì ∈ [0, 1] is the anamorphic interpo- 푥 = 푓푥 푦 (8b) 휑 ì2 ì2 ì2 ì2 ì푦 푓푥 +푓푦 푓푥 +푓푦 lation weight, defined in Section 2 on page 2. 푟 2 푓ì2 푓ì2 = 푥 + 푦 (8c) " # " # !−1 5 Anamorphic chromatic aberration 푣 ′ 푓ì 푘ì 푟 2 푘ì 푟 4 휑 ì푥 푥 1 + 푥1 + 푥2 ··· ì푥 ′ = · □ A chromatic aberration effect can be achieved with multi- 푣ì 푓ì + 푘ì 푟 2 + 푘ì 푟 4 ··· 휑ì푦 푦 푦 1 푦1 푦2 sample blur, where each sample layer is colored with the | {z } corresponding spectral-value [Gilcher 2015]. Presented pe- radial anamorphic 휒ì " # " # ! (8d) riodic function for spectral color produces samples that 푓ì 푓ì 푝ì 푞ì 푐ì always add up to 1 (neutral white) when their number is + 푥 푥 · 1 + 푟 2 1 + 1 ì ì 푝 ì 푞 푐 □ 푓푦 푓푦 2 ì2 ì2 even. It also presents the correct order of spectrum colors. | {z } | {z } |{z} decentering thin prism cardinal b 휒ì푟  1/4 ! 퐺ˆ   1 3   푣 ′  푥  휒ì = clamp − 4 mod 푡 + 0 , 1 − 2 , (9) ì푥    푔   ′ ↦→ 퐺ˆ푦 , □ (8e) 휒  0 2 3  푣ì    ì푏  /4 푦 퐺ˆ       푧  where 휒ì ∈ [0, 1]3 is the spectral-color value for position where 푐ì1, 푐ì2 are the cardinal-offset parameters, 푞ì1, 푞ì2 are 푡 ∈ [0, 1]. See Figure 4 on the next page for more information. the thin-prism distortion parameters and 푝ì1, 푝ì2 are the de- Performing a spectral blur on an image involves the multi- centering parameters. A set of 푘ì parameters define radial sample sum of spectrum-colored layers. Here 푡 (replaced by distortion for each anamorphic power axis. 푣ì is the input sample progress) never reaches 1, which preserves picture view-coordinate, and 푣ì′ is the view coordinate with applied white-balance. The number of samples 푛 must be an even Fober, J.M.

0 1

Figure 4. Graph mapping of 푡 ∈ [0, 1] to spectral color Figure 5. Example of anamorphic lens distortion with chro- 휒 , 3 ì ì ì ∈ [0 1] , for simulation of chromatic aberration. This is matic aberration, where 푘푥1 = −0.25, 푘푦1 = 0.04, 푑 = 0.5, an output of periodic function found in Equation (9). The with 64–spectral levels. distribution of values ensures proper color order and sum- of-samples with a neutral-white tint. be performed, in which the direction is perpendicular to 푠ì vector, and with smaller magnitude, as below. number, no less than 2 for a correct result. 푠 ′ 푠 ì푥 1 −ì푦 푓ì′ 푓ì ′ = , (13)  푟  푛−1  푟  1/4 ! 푠ì 푠ì   2 ∑︁   1 3 푖   푦 4 푥 푓ì′ 푓ì − +   , − , 푔 = 푔 clamp 4 mod  0  1 2 푠 ′   푛   0 2 푛   where ì is the second-pass blur direction vector.   푖=0   3/4 푓ì′ 푓ì    푏   푏    Figure 5 presents the final effect of two-pass blur, where | {z } 푣ì 휒ì periodic function the first pass is spectral, along lens-distortion Δ , and the (10) second-pass of quarter-magnitude is perpendicular ⊥Δ푣ì.A where 푛 ∈ 2N1 is the even number of samples for the chro- combination of both adds a defocussing effect to the distor- matic aberration color-split. 푓ì ∈ [0, 1]3 is the current-sample tion. 푓ì′ ∈ [ , ]3 position color-value. 0 1 is the final spectral-blurred 6 Final thoughts color value. The equation for spectral color 휒ì can be rewritten to a In this article, a mathematical model for asymmetrical anamorphic- more computationally optimized form. perspective geometry (pantomorphic) has been presented, 1 1 along with perception-driven distortion design recommenda- 휒 clamp 3/2 − |4푡 − 1| clamp 4푡 − 7/2  ì푟   0   0  tions. Each power axis of the pantomorphic model resembles    1    휒ì  3/2 − | 푡 − |+  ,  푔 =  clamp0 4 2   0  parametric azimuthal projection. Such parametrization en- 휒   1   1   ì푏  − clamp 3/2 − |4푡 − 1| 1 − clamp 4푡 − 7/2 ables a picture’s geometry to be adaptive, which can dynam-    0   0     (11) ically adjust to the visible content in an artistically convinc- where 휒ì ∈ [0, 1]3 is the spectral color at position 푡 ∈ [0, 1]. ing manner. Along with anamorphic perspective, vignetting See Figure 4 for visualization. and lens-distortion with integrated chromatic-aberration has been provided. Selection of these features enables holistic 5.1 Chromatic aberration through lens distortion digital-lens simulation for immersive-imagery production. Chromatic aberration can be integrated into lens distortion, with spectral blurring, through lens-transformation vector References Δ푣ì. Below, the equation for spectral-blur displacement vector Leon B. Alberti. 1435. On Painting (1970 ed.). New Haven: Yale University 푠ì is presented, calculated per the sample at position 푡. Press. http://www.noteaccess.com/Texts/Alberti/index.htm Giulio C. Argan and Nesca A. Robb. 1946. The Architecture of Brunelleschi 푣 푣 ′ 푣 Δì푥 ì푥 − ì푥 and the Origins of Perspective Theory in the Fifteenth Century. Journal 푣 = 푣 ′ 푣 (12a) Δì푦 ì푦 − ì푦 of the Warburg and Courtauld Institutes 9 (1946), 96. https://doi.org/10. 2307/750311 푠ì 1 Δ푣ì 푥 = 1 + 푡 − 푑 푥 , (12b) Joseph Baldwin, Alistair Burleigh, and Robert Pepperell. 2014. Comparing 푠ì푦 2 Δ푣ì푦 Artistic and Geometrical Perspective Depictions of Space in the Visual 푠ì ∈ R2 Field. i-Perception 5, 6 (jan 2014), 536–547. https://doi.org/10.1068/i0668 where is the spectral blur sample-offset vector at Felix Bettonvil. 2005. Fisheye lenses. WGN, Journal of the International position 푡 ∈ [0, 1). Value 푑 ∈ R denotes the lens dispersion- Meteor Organization 33, 1 (2005), 11–12. https://ui.adsabs.harvard.edu/ scale. For a visually pleasing result, additional blur pass can link_gateway/2005JIMO...33....9B/ADS_PDF Pantomorphic Perspective for Immersive Imagery

Panorama source: captured from Conan Exiles through Nvidia Panorama source: Grzegorz Wronkowski, under CC-BY license. Ansel. (b) 푘ì = [−1/2 0], 푓 = 1, Ω = 120°, flying. ì ℎ (a) 푘 = [1/2 − 1/2 0], 푓 = 0.618, Ωℎ ≈ 156°, (asymmetrical) racing.

Panorama source: captured from Obduction through Nvidia Ansel. Panorama source: captured from For Honor through Nvidia Ansel. ì ì (c) 푘 = [0 3/4 − 1/2], 푓 = 0.82, Ωℎ ≈ 140°, (asymmet- (d) 푘 = [0 − 1/2], 푓 = 0.63, Ωℎ ≈ 182°, stereopsis (cyclo- rical) first-person aim. pean).

Figure 6. Examples of super wide-angle views in anamorphic projection with natural vignetting. Mapped from 360° panorama in equirectangular projection.

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