whitted Raytracing

COS350 - Computer Graphics

Image Formation via raytrace rendering

Raytracing

a specific rendering algorithm that simulates light interacting with scene (geometry+materials) and enterring camera

Raytracing algorithm

Use Whitted-style ray tracing algorithm to generate image

• given: camera, geometry + materials, lights
• determine visibility and compute colors

for each pixel {
    determine viewing direction
    intersect ray with scene
    compute illumination
    store result in pixel
}

Viewing

for each pixel {
    determine viewing direction <==
    intersect ray with scene
    compute illumination
    store result in pixel
}

Viewer Model

a painter tracing objects on a canvas in front

Viewer Model

simplify to pinhole photography, move photo film / sensor in front of focal point

Viewer Model – Parameters

• camera position \(\tilde\o\) and orientation \(\hat\x\), \(\hat\y\), \(\hat\z\)
  • NOTE: \(\hat\z\) points backwards; \(-\hat\z\) points fowards
• image plane: distance \(d\) and size \(w\), \(h\)

View frustum

Note: all visible points within a truncated pyramid

Generating view rays

• for each pixel, ray from camera center to the pixel center

Generating view rays

convert pixel coordinates \((x,y)\) to image plane coordinates \((u,v)\)

• digital image coords: \((x,y)\) where \(x \in [0,w-1]\) and \(y \in [0,h-1]\)
• image plane coords: \((u,v) \in [0,1]^2\), origin at bottom-left

\[ u = \frac{x + 0.5}{w} \] \[ v = 1 - \frac{y + 0.5}{h} \]

Generating view rays

compute point \(\point{q}\) at center of current pixel \((x,y)\)

• \(\point{q}(u,v) = \point{o} + (u-0.5) w \direction{x} + (v-0.5) h \direction{y} - d \direction{z}\)

Generating view rays

computer ray that starts at camera focal point \(\point{o}\) and passes through pixel center \(\point{q}\)

• ray: \(\tilde\p(t) = \tilde\e + t \hat\d\), where \(\tilde\e = \tilde\o\) and \(\hat\d = (\tilde\q-\tilde\o) / || \tilde\q-\tilde\o ||\)

Generating view rays

• previous formulations were defined wrt the world
• frames are mathematically equivalent, cleaner, and more elegant
• camera frame: \(\check\f' = \{ \tilde\f'_o, \hat\f'_x, \hat\f'_y, \hat\f'_z \}\), defined wrt to world
• all entities defined wrt to \(\check\f'\), then transformed to world frame

\[ \tilde\o = 0,\quad \hat\x = (1,0,0),\quad \hat\y = (0,1,0),\quad \hat\z = (0,0,1) \]

\[\begin{array}{rcl} \tilde\q(u,v) & = & \tilde\o + (u-0.5) w \hat\x + (v-0.5) h \hat\y - d \hat\z \\ & = & 0 + w (u-0.5) (1,0,0)\, + \\ & & +\, h (v-0.5) (0,1,0) - d (0,0,1) \\ & = & (w * (u-0.5), h * (v-0.5), -d) \end{array}\]

Generating view rays

pixel center:

\[ \begin{array}{rcl} \tilde\q'(u,v) & = & O + (u-0.5) w \hat\x' + (v-0.5) h \hat\y' - d\hat\z' \\ & = & \mat{ (u-0.5)w & (v-0.5)h & -d }^T \end{array} \]

notes:

• \(O = \mat{0 & 0 & 0}^T\) (zero vector)
• \(\hat\x' = \mat{1 & 0 & 0}^T, \hat\y' = \mat{0 & 1 & 0}^T, \hat\z' = \mat{0 & 0 & 1}^T\)

Generating view rays

view ray:

\[ \begin{array}{rcl} \tilde\p'(t) & = & \tilde\e + t \hat\d \\ & = & O + t (\tilde\q - O) / || \tilde\q - O || \\ & = & t \hat\q \end{array} \]

note: last line abuses notation by mixing up points and vectors

Intersection

for each pixel {
    determine viewing direction
    intersect ray with scene    <==
    compute illumination
    store result in pixel
}

Geometry Model

• simple shapes
  • spheres, planes, quads, triangles
• complex shapes
  • handled as collections of simple shapes later in the course

Ray-Shape Intersection

• determine visible surface by finding closest intersection along a ray
• ray \(\r: \tilde\e + t \hat\d\) with \(t \in (t_{min},t_{max})\)
  • keep explicit bounds on \(t\)
  • if not specified otherwise: \(t_{min} = \epsilon\), \(t_{max} = \infty\)
  • e.g. used in shadows and to improve numerical precision
  • \(\epsilon\) mitigate numerical precision issues ("shadow acne")
    • value is scene dependent: start with \(10^{-5}\)

Ray-Sphere Intersection

point on a ray: \(\tilde\p(t) = \tilde\e + t\hat\d\)

point on a sphere: \(|| \tilde\p(t)-\tilde\c || = r\)

by substitution: \( || \tilde\e + t\hat\d - \tilde\c || = r \)

Ray-Sphere Intersection

algebraic equation: \(a t^2 + b t + c = 0\)

with: \(a = ||\hat\d||^2\), \(b=2 \hat\d \* (\tilde\e-\tilde\c)\), \(c = ||\tilde\e-\tilde\c||^2 - r^2\)

determinant: \(d = b^2 - 4ac\)

no solution for \(d < 0\)

Ray-Sphere Intersection

two solutions

\[t_{\pm} = \frac{-b \pm \sqrt{d}}{2a}\,,\quad d=b^2-4ac\]

pick smallest \(t\) such that \(t \in ( t_{min}, t_{max} )\)

Ray-Sphere Intersection

shading frame \(\frame{f} = \{ \point{o}, \direction{x}, \direction{y}, \direction{z} \}\)

• \(\point{o} = \point{p}(t) = \point{e} + t \direction{d}\)
• \(\direction{x} = (\sp, \cp, 0)\)
• \(\direction{y} = (\ct\cdot\cp, \ct\cdot\sp, \st)\)
• \(\direction{z} = (\point{p} - \point{c}) / r\)

where

• \(\theta = \arccos(n_z)\)
• \(\phi = \arctan(n_y, n_x)\)

attributes (ex: color or uv) can be interpolated at \(\point{p}(t)\) using \(\theta\), \(\phi\)

\[ u = 1 - \frac{\phi}{\pi} \qquad v = 0.5 + \frac{\theta}{2 \pi} \]

Ray-Plane Intersection

point on a ray: \(\tilde\p(t) = \tilde\e + t\hat\d\)

point on a plane: \((\tilde\p(t)-\tilde\c) \* \hat\n = 0\)

by substitution: \( (\tilde\e + t\hat\d -\tilde\c) \* \hat\n = 0 \)

Ray-Plane Intersection

one solution for \(\hat\d \* \hat\n \neq 0\), no/infinite solutions otherwise

\[t = \frac{(\tilde\c - \tilde\e) \* \hat\n}{\hat\d \* \hat\n}\]

check that \(t \in ( t_{min}, t_{max} )\)

Ray-Plane Intersection

shading frame \(\frame{f} = \{ \point{o}, \direction{x}, \direction{y}, \direction{z} \}\)

• \(\point{o} = \point{p}(t) = \point{e} + t \direction{d}\)
• \(\direction{x} = \text{surface frame } \direction{x}\)
• \(\direction{y} = \text{surface frame } \direction{y}\)
• \(\direction{z} = \direction{n} = \text{surface frame } \direction{z}\)

Ray-Disk Intersection

Ray-Disk intersection uses Ray-Plane intersection with additional condition

• compute the L² norm of the vector from surface center \(\point{c}\) to intersection \(\point{p}(t)\) as \(||\point{p}(t) - \point{c}||_2\)
• miss if distance is larger than radius; otherwise hit

Ray-Quad Intersection

Ray-Quad intersection uses Ray-Plane intersection with additional condition

• compute the L^∞ norm on the vector from surface center \(\point{c}\) to intersection \(\point{p}(t)\) as \(||\point{p}(t) - \point{c}||_\infty\)
• miss if distance is larger than radius; otherwise hit

to handle rotated quad, compute distance of \(\point{p}(t)\) from origin in the local frame (i.e., transform \(\point{p}(t)\) to local frame, \(\point{p}'(t)\))

attributes (ex: color or uv) can be interpolated at \(\point{p}(t)\) using \(\point{p}'(t)\)

\[ u = 0.5 + \frac{p_x'}{2r} \qquad v = 0.5 + \frac{p_y'}{2 r} \]

Ray-Triangle Intersection

point on ray: \(\tilde\p(t) = \tilde\e + t\hat\d\)

point on triangle: \(\tilde\p(\alpha,\beta) = \alpha(\tilde\a-\tilde\c) + \beta(\tilde\b-\tilde\c) + \tilde\c\)

by substitution: \(\tilde\e + t\hat\d = \alpha(\tilde\a-\tilde\c) + \beta(\tilde\b-\tilde\c) + \tilde\c\)

Ray-Triangle Intersection

\[ \tilde\e + t\hat\d = \alpha(\tilde\a-\tilde\c) + \beta(\tilde\b-\tilde\c) + \tilde\c \rightarrow \]

\[ \alpha(\tilde\a-\tilde\c) + \beta(\tilde\b-\tilde\c) - t\hat\d = \tilde\e - \tilde\c \rightarrow \]

\[ \alpha\a' + \beta\b' - t\hat\d = \e' \rightarrow \]

\[ \mat{{-\hat\d} & \a' & \b'} \mat{ t \\ \alpha \\ \beta } = \e' \]

Ray-Triangle Intersection

use Cramer's rule

\[ t = \dfrac{\matdet{\e' & \a' & \b'}}{\matdet{{-\hat\d} & \a' & \b'}} = \dfrac{(\e' \xx \a') \* \b'}{(\hat\d \xx \b') \* \a'} \]

\[ \alpha = \dfrac{\matdet{{-\hat\d} & \e' & \b'}}{\matdet{{-\hat\d} & \a' & \b'}} = \dfrac{(\hat\d \xx \b') \* \e'}{(\hat\d \xx \b') \* \a'} \]

\[ \beta = \dfrac{\matdet{{-\hat\d} & \a' & \e'}}{\matdet{{-\hat\d} & \a' & \b'}} = \dfrac{(\e' \xx \a') \* \hat\d}{(\hat\d \xx \b') \* \a'} \]

test for

\[ t \in (t_{min},t_{max}), \,\,\,\, \alpha \ge 0, \,\,\,\, \beta \ge 0, \,\,\,\, \alpha + \beta \le 1 \]

Ray-Triangle Intersection

shading frame \(\frame{f} = \{ \point{o}, \direction{x}, \direction{y}, \direction{z} \}\)

• \(\point{o} = \point{p}(t) = \point{e} + t \direction{d}\)
• \(\direction{x} = \dfrac{\point{b} - \point{a}}{||\point{b} - \point{a}||}\)
• \(\direction{z} = \dfrac{\direction{x} \times (\point{c} - \point{a})}{||\direction{x} \times (\point{c} - \point{a}) ||}\)
• \(\direction{y} = \direction{z} \times \direction{x}\)

vertex attributes (ex: color or uv) can be interpolated at \(\point{p}(t)\) using \(\alpha\), \(\beta\), \(\gamma\) as linear weights

\[ c_{\point{p}(t)} = \alpha c_{\point{a}} + \beta c_{\point{b}} + \gamma c_{\point{c}} \]

Intersection and Coord Systems

• transform the object
  • simple for triangles, since they transforms to triangles
  • but objects may require more complex intersection tests
• transform the ray
  • much more elegant
  • works on any surface
  • allow for much simpler intersection tests

Intersection and Coord Systems

• ray \(\r=\{ \tilde\e,\hat\d \}\) wrt \(\check\f\) (e.g. world)
• object \(o'\) defined wrt \(\check\f'\) (in turn defined wrt \(\check\f\))
• transform rays \(\r'=\{\tilde\e',\hat\d'\}\)
  • transform origin/direction as point/vector
• intersect object \(o'\) with transformed ray \(\r'\)
  • use standard intersection tests
• transform intersection frame back to \(\check\f\)
  • transform origin/axes as point/vectors

Image so far

Intersecting many shapes

• intersect each primitive
• pick closest intersection
• essentially a line search

Intersecting many shapes – pseudocode

maxDistance = infinity
hit = null
foreach surface s {
  if(s.intersect(ray,intersection)) {
    if(intersection.distance < maxDistance) {
      hit = s
      maxDistance = intersection.distance
    }
  }
}

Image so far