Ray Tracing Graphics & Visualization: Principles & - PowerPoint PPT Presentation

Graphics & Visualization Chapter 15 Ray Tracing Graphics & Visualization: Principles & Algorithms Chapter 15

Introduction • Direct-rendering algorithms:  Sorting is performed in image space (Z-buffer)  Object-to-screen space image synthesizers • Ray tracing is a general algorithm that operates in the opposite manner:  It is a screen-to-object space image synthesizer • In ray tracing , we follow rays along the line of sight passing through each pixel as they travel through the 3-D scene  registers what the observer sees along this direction • As ray encounters geometric entities:  Specularly reflected, refracted, or attenuated (completely absorbed) 2 Graphics & Visualization: Principles & Algorithms Chapter 15

Introduction (2) • Hidden surface elimination happens as part of this process:  Ray encounters surface interfaces closer to the viewer first while it travels through the 3-D world • Simple direct rendering relies on local shading models:  Shadows & reflected/refracted light on surfaces need to be simulated separately & fused as color information in the local illumination model used during scan conversion • Ray tracing, integrates all calculations that involve specular transmission of light in a single & elegant recursive algorithm, the recursive ray tracing algorithm 3 Graphics & Visualization: Principles & Algorithms Chapter 15

Principles of Ray Tracing • Light transmission:  An infinite number of rays emanate from a light source, and a small number reach the eye after following complex paths within the scene • Light is diffusely scattered and specularly reflected or refracted • Part of the specularly & diffusely reflected light is directly received by the observer 4 Graphics & Visualization: Principles & Algorithms Chapter 15

Principles of Ray Tracing (2) • Light reaches the observer indirectly as well:  Following paths through transparent media or by being reflected off perfect mirrors • But:  Given a very large but finite number of rays starting from the light sources, it is statistically unlikely that they will hit the image pixels and contribute to the final result  We must find a different sampling mechanism that ensures that the image pixels are crossed by the light paths and thus adequately sampled 5 Graphics & Visualization: Principles & Algorithms Chapter 15

Principles of Ray Tracing (3) • In ray tracing: Light seen through a pixel of the rendered image is the cumulative contribution of the rays that directly or indirectly hit the surface point visible in this direction and that travel toward the viewpoint • Nearest point encountered by looking at the scene though pixel ( i , j ) in general obstructs all other geometry behind it:  Point may or may not be directly illuminated by the light source(s), depending on whether other geometry prevents the light from reaching it  Paths reaching intersection point from other directions traveling toward pixel ( i , j ) can be followed to discover what light they contribute • This is possible due to the reciprocity of light propagation:  Light follows same path during refraction or perfect reflection on a material interface regardless of the direction of propagation 6 Graphics & Visualization: Principles & Algorithms Chapter 15

Principles of Ray Tracing (4) • If it is directly lit by the light source  local illumination model can be applied • Cumulative illumination visible through a frame-buffer pixel ( i , j ) due to the contribution of direct and indirect rays: 7 Graphics & Visualization: Principles & Algorithms Chapter 15

Principles of Ray Tracing (5) • Since we are only interested in those rays that eventually reach the viewpoint through a viewport pixel  Trace back the light contributions by following the rays in the opposite direction of their propagation toward the source • For each ray back to its source  evaluate the light propagated toward the viewer by applying a local illumination model & re- investigating for other secondary rays that reach that point:  This is exactly the mechanism of ray tracing  A computationally manageable problem 8 Graphics & Visualization: Principles & Algorithms Chapter 15

Principles of Ray Tracing (6) • Compared to direct-rendering algorithms, ray tracing has 2 significant advantages :  Ray-geometry intersections can be directly performed using non- polygonal surfaces, such as geometric solids, implicit or parametric surfaces, and fractals, without requiring any conversion to polygons first  Mathematical surfaces that can be intersected by a ray can be rendered  Reflection & refraction phenomena can be accurately modeled 9 Graphics & Visualization: Principles & Algorithms Chapter 15

Ray Diversion Reflection Refraction 10 Graphics & Visualization: Principles & Algorithms Chapter 15

Reflection ˆ i • For an arbitrary ray of light from a direction incident on a r perfectly reflecting interface between 2 bodies, the reflected ray ˆ r in the perfect mirror-reflection direction is: r (15.1)   ˆ ˆ ˆ ˆ ˆ 2 ( · ) r r n n r r i i • Notice that here the incident direction is the opposite of the light ˆ direction vector since we need to emphasize the direction of l propagation for clarity 11 Graphics & Visualization: Principles & Algorithms Chapter 15

Refraction • Simple index of refraction (or refractive index ) n : Ratio between speed of light c in a vacuum & the phase velocity of light u in this medium:  / n c u (15.2) • n > 1: Transparent materials & n ~ 1: Air • n depends on wavelength λ of the light  n = n ( λ )  For visible light: n ↓ when wavelength  • Phase velocity is responsible for the bending of the propagation direction as the light crosses the interface between them  sin n • According to Snell's law : (15.3)   1 t sin n 2 i  Light entering a medium with larger index of refraction ( n 2 > n 1 ) is bent toward the normal direction of the optically denser medium 12 Graphics & Visualization: Principles & Algorithms Chapter 15

Refraction (2) • When ( n 2 < n 1 ) total internal reflection may occur  Light not transmitted through the boundary but reflected back • Min angle of incidence at which total internal reflection occurs is called a critical angle θ c :   n    2  arcsin (15.4) c   n 1 13 Graphics & Visualization: Principles & Algorithms Chapter 15

Refraction (3) • Calculation of the direction of the new, transmitted ray :      ˆ ˆ ˆ cos si n , (15.5) r n g t t t ˆ ˆ p g r where : unit length vector || and:            ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ cos ·( · ) ( ) · r r n r n r n r n r n (15.6) p i i i i i i ˆ i   ˆ r • is a unit vector  length of si n r p i • After normalizing it, we get: ˆ   ˆ ˆ ˆ ˆ r ( · ) r n n r (15.7)   ˆ p i i g   sin s in i i ˆ g • Replacing from (15.7) into (15.5):  sin   (15.8)  ˆ ˆ ˆ ˆ ˆ ˆ t =- cos - ( · )- . r n n nr r  t t i sin i 14 Graphics & Visualization: Principles & Algorithms Chapter 15

Refraction (4) • From (15.3) we can replace the sines in the above relation with the indices of refraction     • Also, using Pythagorean trigonometric identity 2 cos 1 sin t t we get: 2 2   n n            2 2 2 (15.9) 1 1 cos 1 sin 1 sin 1 1 cos t t i i 2 2 n n 2 2 • Introducing these relations in (15.8): 2     n n          ˆ ˆ ˆ ˆ ˆ ˆ 2 1 1 1 1 cos · r n n nr r t i i i 2 (15.10) n n 2 2 • Final step  replace cosine with corresponding inner product:       2 2   n n n n n             2        2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ  ˆ ˆ ˆ ˆ  (15.11) 1 1 1 1 1 1 1 · · · 1 1 · r n n r n n r r r n n r n r   t 2 i i i i i 2 i n n n n n   2 2 2 2 2 15 Graphics & Visualization: Principles & Algorithms Chapter 15

Refraction (5) • Note that quantity inside the radical of (15.11) is > 0:  If < 0  total internal refraction & new ray is calculated from (15.1) • Refracted ray calculation: 16 Graphics & Visualization: Principles & Algorithms Chapter 15