INFOMAGR – Advanced Graphics Jacco Bikker - November 2019 - February 2020 Lecture 2 - “ Whitted ” Welcome! 𝑱 𝒚, 𝒚 ′ = 𝒉(𝒚, 𝒚 ′ ) 𝝑 𝒚, 𝒚 ′ + න 𝝇 𝒚, 𝒚 ′ , 𝒚 ′′ 𝑱 𝒚 ′ , 𝒚 ′′ 𝒆𝒚′′ 𝑻
Today’s Agenda: ▪ Introduction: Appel ▪ Whitted ▪ Cook ▪ Assignment 1
Advanced Graphics – Whitted 3 Appel Some Techniques for Shading Machine Renderings of Solids, Arthur Appel, 1964. Idea: use rays to find geometry and shadows.
Advanced Graphics – Whitted 4 Appel Plane: 𝑄 ∙ 𝑂 + 𝑒 = 0 Ray: 𝑄(𝑢) = 𝑃 + 𝑢𝐸 Recap Substituting for 𝑄(𝑢) , we get 𝑃 + 𝑢𝐸 ∙ 𝑂 + 𝑒 = 0 𝑄 1 𝑢 = −(𝑃 ∙ 𝑂 + 𝑒)/(𝐸 ∙ 𝑂) u 𝑄 0 𝑄 = 𝑃 + 𝑢𝐸 v 𝐹 𝑄 2
Advanced Graphics – Whitted 5 Appel Ray: 𝑄(𝑢) = 𝑃 + 𝑢𝐸 𝒖
Advanced Graphics – Whitted 6 Appel
Today’s Agenda: ▪ Introduction: Appel ▪ Whitted ▪ Cook ▪ Assignment 1
Advanced Graphics – Whitted 8 Whitted An Improved Illumination Model for Shaded Display In 1980, “State of the Art” consisted of: ▪ Rasterization ▪ Shading: either diffuse (N · L) or specular ((N · H) n ), both not taking into account fall-off (Phong) ▪ Reflection, using environment maps (Blinn & Newell *) ▪ Stencil shadows (Williams **) * : Blinn, J. and Newell, M. 1976. Texture and Reflection in Computer Generated Images. ** : Williams, L. 1978. Casting curved shadows on curved surfaces..
Advanced Graphics – Whitted 9 Whitted An Improved Illumination Model for Shaded Display In 1980, “State of the Art” consisted of: ▪ Rasterization ▪ Shading: either diffuse (N · L) or specular ((N · H) n ), both not taking into account fall-off (Phong) ▪ Reflection, using environment maps (Blinn & Newell) ▪ Stencil shadows (Williams) Goal: ▪ Solve reflection and refraction Improved model: ▪ Based on classical ray optics
Advanced Graphics – Whitted 10 Whitted An Improved Illumination Model for Shaded Display* Physical basis of Whitted-style ray tracing: Light paths are generated (backwards) from the camera to the light sources, using rays to simulate optics. Color Trace( ray r ) I, N, mat = NearestIntersection( scene, r ) return mat.color * DirectIllumination( I, N ) * : T. Whitted. An Improved Illumination Model for Shaded Display. Commun. ACM, 23(6):343 – 349, 1980.
Advanced Graphics – Whitted 11 Whitted An Improved Illumination Model for Shaded Display Color Trace( ray r ) I, 𝑂 , mat = NearestIntersection( scene, r ) return mat.color * DirectIllumination( I, 𝑂 ) Direct illumination: Summed contribution of unoccluded point light sources, taking into account: ▪ Distance to I ▪ Angle between 𝑀 and 𝑂 ▪ Intensity of light source Note that this requires a ray per light source.
Advanced Graphics – Whitted 12 Whitted An Improved Illumination Model for Shaded Display Color Trace( ray r ) I, 𝑂 , mat = NearestIntersection( scene, r ) if (mat == DIFFUSE) return mat.color * DirectIllumination( I, 𝑂 ) if (mat == MIRROR) return mat.color * Trace( I, reflect( r. 𝐸 , 𝑂 ) Indirect illumination: For perfect specular object (mirrors) we extend the primary ray with an extension ray: ▪ We still modulate transport with the material color ▪ We do not apply 𝑂 ∙ 𝑀 ▪ We do not calculate direct illumination
Advanced Graphics – Whitted 13 Whitted Reflection Given a ray direction 𝐸 and a normalized surface normal 𝑂 , the reflected vector 𝑆 = 𝐸 − 2(𝐸 ∙ 𝑂)𝑂 . Derivation: 𝑊 = 𝑂(𝐸 ∙ 𝑂) 𝑆 𝐸 𝑉 𝑉 = 𝐸 − Ԧ 𝑤 𝑆 = 𝑉 + (−𝑊) 𝑶 𝑆 = 𝐸 − 𝑂 𝐸 ∙ 𝑂 − 𝑂(𝐸 ∙ 𝑂) 𝑊 𝑆 = 𝐸 − 2(𝐸 ∙ 𝑂)𝑂
Advanced Graphics – Whitted 14 Whitted Question 1: For direct illumination, we take into account: ▪ Material color ▪ Distance to light source ▪ 𝑂 ∙ 𝑀 Why? Question 2: We use the summed contribution of all light sources. Is this correct? Question 3: Why do we not sample the light sources for a pure specular surface? (can you cast a shadow on a bathroom mirror?) Question 4: Prove geometrically that, for normalized vectors 𝐸 and 𝑂 , 𝑆 = 𝐸 − 2 𝐸 ∙ 𝑂 𝑂 yields a normalized vector.
Advanced Graphics – Whitted 15 Whitted An Improved Illumination Model for Shaded Display Handling partially reflective materials: Color Trace( ray r ) I, 𝑂 , mat = NearestIntersection( scene, r ) s = mat.specularity d = 1 – mat.specularity return mat.color * ( s * Trace( ray( I, reflect( r. 𝐸 , 𝑂 ) ) ) + d * DirectIllumination( I, 𝑂 ) ) Note: this is not efficient. (why not?)
Advanced Graphics – Whitted 16 Whitted An Improved Illumination Model for Shaded Display Dielectrics Color Trace( ray r ) I, 𝑂 , mat = NearestIntersection( scene, r ) if (mat == DIFFUSE) return mat.color * DirectIllumination( I, 𝑂 ) if (mat == MIRROR) return mat.color * Trace( I, reflect( r. 𝐸 , 𝑂 ) if (mat == GLASS) return mat.color * ?
Advanced Graphics – Whitted 17 Whitted Dielectrics 𝑶 𝐸 The direction of the transmitted vector 𝑈 depends on 𝜄 1 the refraction indices 𝑜 1 , 𝑜 2 of the media separated by the surface. According to Snell’s Law: 𝑜 1 𝑜 1 𝑡𝑗𝑜𝜄 1 = 𝑜 2 𝑡𝑗𝑜𝜄 2 𝑜 2 or 𝑜 1 𝜄 2 𝑡𝑗𝑜𝜄 1 = 𝑡𝑗𝑜𝜄 2 𝑜 2 𝑈 Note: left term may exceed 1, in which case 𝜄 2 cannot be computed. Therefore: 𝑜 2 𝑡𝑗𝑜𝜄 1 = 𝑡𝑗𝑜𝜄 2 ⟺ 𝑡𝑗𝑜𝜄 1 ≤ 𝑜2 𝑜 1 𝑜 2 𝑜1 ➔ 𝜄 𝑑𝑠𝑗𝑢𝑗𝑑𝑏𝑚 = arcsin 𝑜 1 sin 𝜄 2
Advanced Graphics – Whitted 18 Whitted Dielectrics 𝑶 𝐸 𝑜 1 𝑜2 𝑜 2 𝑡𝑗𝑜𝜄 1 = 𝑡𝑗𝑜𝜄 2 ⟺ 𝑡𝑗𝑜𝜄 1 ≤ 𝜄 1 𝑜1 𝑜 1 2 𝑜 1 2 𝑙 = 1 − 1 − 𝑑𝑝𝑡𝜄 1 𝑜 2 𝑜 2 𝑈𝐽𝑆, 𝑔𝑝𝑠 𝑙 < 0 𝜄 2 𝑜 1 𝐸 + 𝑂 𝑜 1 𝑈 = ൞ 𝑈 𝑑𝑝𝑡𝜄 1 − 𝑙 , 𝑔𝑝𝑠 𝑙 ≥ 0 𝑜 2 𝑜 2 Note: 𝑑𝑝𝑡𝜄 1 = 𝑂 ∙ −𝐸 , and 𝑜 1 𝑜 2 should be calculated only once. * For a full derivation, see http://www.flipcode.com/archives/reflection_transmission.pdf
Advanced Graphics – Whitted 19 Whitted Dielectrics 𝑆 𝑶 𝐸 A typical dielectric transmits and reflects light. 𝑈
Advanced Graphics – Whitted 20 Whitted Dielectrics A typical dielectric transmits and reflects light. Based on the Fresnel equations, the reflectivity of the surface for non-polarized light is formulated as: 2 2 𝑠 = 1 𝑜 1 𝑑𝑝𝑡𝜄 𝑗 − 𝑜 2 cos 𝜄 𝑢 + 𝑜 1 cos 𝜄 𝑢 − 𝑜 2 cos 𝜄 𝑗 𝐺 2 𝑜 1 𝑑𝑝𝑡𝜄 𝑗 + 𝑜 2 cos 𝜄 𝑢 𝑜 1 cos 𝜄 𝑢 + 𝑜 2 cos 𝜄 𝑗 Reflectance for s-polarized light Reflectance for p-polarized light Reflectance for unpolarized light 2 𝑜 1 Where: cos 𝜄 𝑢 = 1 − 𝑜 2 sin 𝜄 𝑗
Advanced Graphics – Whitted 21 Whitted Dielectrics 𝑆 𝑶 𝐸 𝐺 𝑠 = ⋯ Based on the law of conservation of energy: 𝐺 𝑢 = 1 − 𝐺 𝑠 𝑈
Advanced Graphics – Whitted 22 Whitted Ray Tracing World space ▪ Geometry ▪ Eye ▪ Screen plane ▪ Screen pixels ▪ Primary rays ▪ Intersections ▪ Point light ▪ Shadow rays Light transport ▪ Extension rays Light transport
Advanced Graphics – Whitted 23 Whitted Ray Tree Using Whitted-style ray tracing, hitting a surface point may spawn: ▪ a shadow ray for each light source; ▪ a reflection ray; ▪ a ray transmitted into the material. The reflected and transmitted rays may hit another object with the same material. ➔ A single primary ray may lead to a very large number of ray queries.
Advanced Graphics – Whitted 24 Whitted Question 5: imagine a scene with several point lights and dielectric materials. Considering the law of conservation of energy, what can you say about the energy transported by each individual ray?
Advanced Graphics – Whitted 25 Whitted Beer’s Law
Advanced Graphics – Whitted 26 Whitted Beer’s Law Light travelling through a medium loses intensity due to absorption. The intensity 𝐽(𝑒) that remains after travelling 𝑒 units through a substance with absorption 𝑏 is: 𝐽 𝑒 = 𝐽 0 𝑓 − ln 𝑏 𝑒 In pseudocode: I.r *= exp( -a.r * d ); I.g *= exp( -a.g * d ); I.b *= exp( -a.b * d );
Advanced Graphics – Whitted 27 Whitted Whitted - Summary A Whitted-style ray tracer implements the following optical phenomena: ▪ Direct illumination of multiple light sources, taking into account Visibility Distance attenuation A shading model: N dot L for diffuse ▪ Pure specular reflections, with recursion ▪ Dielectrics, with Fresnel, with recursion ▪ Beer’s Law The ray tracer supports any primitive for which a ray/primitive intersection can be determined.
Today’s Agenda: ▪ Introduction: Appel ▪ Whitted ▪ Cook ▪ Assignment 1
Advanced Graphics – Whitted 29 Cook ’Distributed Ray Tracing’ Whitted-style ray tracing does not handle glossy reflections, depth of field, motion blur.
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