INFOGR – Computer Graphics Jacco Bikker - April-July 2016 - Lecture 5: “Ray Tracing (3)” Welcome!
Today’s Agenda: Reflections Refraction Recursion Shading models TODO
INFOGR – Lecture 5 – “Ray Tracing (3)” 3 Reflections
INFOGR – Lecture 5 – “Ray Tracing (3)” 4 Reflections Light Transport We introduce a pure specular object in the scene. Based on the normal at the primary intersection point, we calculate a new direction. We follow the path using a secondary ray . At the primary intersection point, we ‘see’ what the secondary ray ‘sees’; i.e. the secondary ray behaves like a primary ray. We still need a shadow ray at the new intersection point to establish light transport.
INFOGR – Lecture 5 – “Ray Tracing (3)” 5 Reflections Reflected Vector Reflection: angle of incidence equals angle of reflection . Using normalized vector 𝐸 : 𝐵 = 𝐸 ∙ 𝑂 𝑂 𝐶 = 𝐸 − 𝐵 𝑆 = 𝐶 + (− 𝑶 𝐵) 𝑆 = 𝐸 − 𝑂 𝐸 ∙ 𝑂 − 𝑂(𝐸 ∙ 𝑂) 𝑆 𝐸 𝑺 = 𝑬 − 𝟑(𝑬 ∙ 𝑶)𝑶 𝛽 𝑗 𝛽 𝑝
INFOGR – Lecture 5 – “Ray Tracing (3)” 6 Reflections Light Transport For a pure specular reflection, the energy from a single direction is reflected into a single outgoing direction. We do not apply 𝑂 ∙ 𝑀 We do apply absorption Since the reflection ray requires the same functionality as a primary ray, it helps to implement this recursively. vec3 Trace( Ray ray ) { I, N, material = scene.GetIntersection( ray ); if (material.isMirror) return material.color * Trace( … ); return DirectIllumination() * material.color; }
INFOGR – Lecture 5 – “Ray Tracing (3)” 7 Reflections What is the color of a bathroom mirror? (In a ray tracer, it’s white.)
INFOGR – Lecture 5 – “Ray Tracing (3)” 8 Reflections Light Transport For pure specular reflections we do not cast a shadow ray. Reason: Light arriving from that direction cannot leave in the direction of the camera.
INFOGR – Lecture 5 – “Ray Tracing (3)” 9 Reflections
INFOGR – Lecture 5 – “Ray Tracing (3)” 10 Reflections Partially Reflective Surfaces We can combine pure specularity and diffuse properties. Situation: our material is only 50% reflective. In this case, we send out the reflected ray, and multiply its yield by 0.5. We also send out a shadow ray to get direct illumination, and multiply the received light by 0.5.
INFOGR – Lecture 5 – “Ray Tracing (3)” 11 Reflections Reflecting a HDR Sky A dark object can be quite bright when reflecting something bright. E.g., a bowling ball, pure specular, color = (0.01, 0.01, 0.01); reflecting a ‘sun’ stored in a HDR skydome, color = (100, 100, 100). For a collection of HDR probes, visit Paul Debevec’s page: http://www.pauldebevec.com/Probes
Today’s Agenda: Reflections Refraction Recursion Shading models TODO
INFOGR – Lecture 5 – “Ray Tracing (3)” 13 Refraction Dielectrics Materials such as water and glass require two types of light transport: 1. Transmission 2. Reflection Both of these happen at each medium boundary .
INFOGR – Lecture 5 – “Ray Tracing (3)” 14 Refraction 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 1 𝑜 2 𝑡𝑗𝑜𝜄 1 = 𝑡𝑗𝑜𝜄 2 ⟺ 𝑡𝑗𝑜𝜄 1 ≤ 𝑜2 𝑜 2 𝑜1 𝜄 𝑑𝑠𝑗𝑢𝑗𝑑𝑏𝑚 = arcsin 𝑜 1 sin 𝜄 2
INFOGR – Lecture 5 – “Ray Tracing (3)” 15 Refraction 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 = 𝑈 𝑑𝑝𝑡𝜄 1 − 𝑙 , 𝑔𝑝𝑠 𝑙 ≥ 0 𝑜 2 𝑜 2 𝑜 1 Note: 𝑑𝑝𝑡𝜄 1 = 𝑂 ∙ −𝐸 , and 𝑜 2 should be calculated only once. * For a full derivation, see http://www.flipcode.com/archives/reflection_transmission.pdf
INFOGR – Lecture 5 – “Ray Tracing (3)” 16 Refraction Dielectrics 𝑶 𝑆 𝐸 A typical dielectric transmits and reflects light. 𝑈
INFOGR – Lecture 5 – “Ray Tracing (3)” 17 Refraction 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 1 − 𝑜 1 1 − 𝑜 1 𝑜 1 𝑑𝑝𝑡𝜄 𝑗 − 𝑜 2 𝑜 2 𝑡𝑗𝑜𝜄 𝑗 𝑜 1 𝑑𝑝𝑡𝜄 𝑗 − 𝑜 2 𝑜 2 𝑡𝑗𝑜𝜄 𝑗 𝑠 = 1 𝐺 + 2 2 2 1 − 𝑜 1 1 − 𝑜 1 𝑜 1 𝑑𝑝𝑡𝜄 𝑗 + 𝑜 2 𝑜 2 𝑡𝑗𝑜𝜄 𝑗 𝑜 1 𝑑𝑝𝑡𝜄 𝑗 + 𝑜 2 𝑜 2 𝑡𝑗𝑜𝜄 𝑗
INFOGR – Lecture 5 – “Ray Tracing (3)” 18 Refraction Dielectrics 𝑶 𝑆 𝐸 In practice, we use Schlick’s approximation: 2 𝑜 1 −𝑜 2 𝑠 = 𝑆 0 + (1 − 𝑆 0 )(1 − 𝑑𝑝𝑡𝜄) 5 , where 𝑆 0 = 𝐺 . 𝑜 1 +𝑜 2 Based on the law of conservation of energy: 𝑈 𝐺 𝑢 = 1 − 𝐺 𝑠
Today’s Agenda: Reflections Refraction Recursion Shading models TODO
INFOGR – Lecture 5 – “Ray Tracing (3)” 20 Recursion Whitted-style Ray Tracing, Pseudocode Todo: Color Trace( vec3 O, vec3 D ) Implement IntersectScene { Implement IsVisible I, N, mat = IntersectScene( O, D ); if (!I) return BLACK; return DirectIllumination( I, N ) * mat.diffuseColor; } Color DirectIllumination( vec3 I, vec3 N ) { vec3 L = lightPos – I; float dist = length( L ); L *= (1.0f / dist); if (!IsVisibile( I, L, dist )) return BLACK; float attenuation = 1 / (dist * dist); return lightColor * dot( N, L ) * attenuation; }
INFOGR – Lecture 5 – “Ray Tracing (3)” 21 Recursion Whitted-style Ray Tracing, Pseudocode Todo: Color Trace( vec3 O, vec3 D ) Handle partially { reflective surfaces. I, N, mat = IntersectScene( O, D ); if (!I) return BLACK; if (mat.isMirror()) { return Trace( I, reflect( D, N ) ) * mat.diffuseColor; } else { return DirectIllumination( I, N ) * mat.diffuseColor; } }
INFOGR – Lecture 5 – “Ray Tracing (3)” 22 Recursion Whitted-style Ray Tracing, Pseudocode Todo: Color Trace( vec3 O, vec3 D ) Implement reflect { Implement refract I, N, mat = IntersectScene( O, D ); Implement Fresnel if (!I) return BLACK; Cap recursion if (mat.isMirror()) { return Trace( I, reflect( D, N ) ) * mat.diffuseColor; } else if (mat.IsDielectric()) { f = Fresnel( … ); return (f * Trace( I, reflect( D, N ) ) + (1- f) * Trace( I, refract( D, N, … ) ) ) * mat.DiffuseColor; } else { return DirectIllumination( I, N ) * mat.diffuseColor; } }
INFOGR – Lecture 5 – “Ray Tracing (3)” 23 Recursion Spheres: pure specular
INFOGR – Lecture 5 – “Ray Tracing (3)” 24 Recursion Spheres: 50% specular
INFOGR – Lecture 5 – “Ray Tracing (3)” 25 Recursion Spheres: one 50% specular, one glass sphere
INFOGR – Lecture 5 – “Ray Tracing (3)” 26 Recursion
INFOGR – Lecture 5 – “Ray Tracing (3)” 27 Recursion Ray Tree Recursion, multiple light sampling and path splitting in a Whitted-style ray tracer leads to a structure that we refer to as the ray tree . All energy is ultimately transported by a single primary ray. Since the energy does not increase deeper in the tree (on the contrary), the average amount of energy transported by rays decreases with depth.
Today’s Agenda: Reflections Refraction Recursion Shading models TODO
INFOGR – Lecture 5 – “Ray Tracing (3)” 29 Shading A diffuse material Diffuse Material appears the same regardless of eye A diffuse material scatters incoming light in all directions. position. 1 Incoming: 𝐹 𝑚𝑗ℎ𝑢 ∗ 𝑒𝑗𝑡𝑢 2 ∗ 𝑂 ∙ 𝑀 Absorption: 𝐷 𝑛𝑏𝑢𝑓𝑠𝑗𝑏𝑚 Reflection: (𝑊 ∙ 𝑂) terms cancel out. 1 Eye sees: (𝑊∙𝑂) 𝑜
INFOGR – Lecture 5 – “Ray Tracing (3)” 30 Shading Specular Material A specular material reflects light from a particular direction in a single outgoing direction. 𝑂
INFOGR – Lecture 5 – “Ray Tracing (3)” 31 Shading
INFOGR – Lecture 5 – “Ray Tracing (3)” 32 Shading Glossy Material A glossy material reflects most light along the reflected vector. 𝑺 = 𝑴 − 𝟑(𝑴 ∙ 𝑶)𝑶 𝑆 For other directions, the amount of energy is: 𝛽 , where exponent 𝛽 determines 𝑊 ∙ 𝑆 the specularity of the surface. 𝑀 𝑾 𝑂
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