173 lines
6.4 KiB
Markdown
173 lines
6.4 KiB
Markdown
---
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geometry: margin=2cm
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output: pdf_document
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---
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# Raytracer part B
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This project implements a raytracer with Blinn-Phong illumination and shadows
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implemented. The primary formula that is used by this implementation is:
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\begin{equation}
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I_{\lambda} =
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k_a O_{d\lambda} +
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\sum_{i=1}^{n_\textrm{lights}} \left(
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f_\textrm{att} \cdot
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S_i \cdot
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IL_{i\lambda} \left[
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k_d O_{d\lambda} \max ( 0, \vec{N} \cdot \vec{L_i} ) +
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k_s O_{s\lambda} \max ( 0, \vec{N} \cdot \vec{H_i} )^n
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\right]
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\right)
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\end{equation}
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Where:
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- $I_{\lambda}$ is the final illumination of the pixel on an object
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- $k_a$ is the material's ambient reflectivity
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- $k_d$ is the material's diffuse reflectivity
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- $k_s$ is the material's specular reflectivity
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- $n_\textrm{lights}$ is the number of lights
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- $f_\textrm{att}$ is the light attenuation factor (1.0 if attenuation is not on)
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- $S_i$ is the shadow coefficient for light $i$
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- $IL_{i\lambda}$ is the intensity of light $i$
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- $O_{d\lambda}$ is the object's diffuse color
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- $O_{s\lambda}$ is the object's specular color
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- $\vec{N}$ is the normal vector to the object's surface
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- $\vec{L_i}$ is the direction from the intersection point to the light $i$
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- $\vec{H_i}$ is halfway between the direction to the light $i$ and the
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direction to the viewer
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- $n$ is the exponent for the specular component
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In this report we will look through how these various factors influence the
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rendering of the scene. All the images along with their source `.txt` files,
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rendered `.ppm` files, and converted `.png` files can be found in the `examples`
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directory of this handin.
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## Varying $k_a$
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$k_a$ is the strength of ambient light. It's used as a coefficient for the
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object's diffuse color, which keeps a constant value independent of the
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positions of the object, light, and the viewer. In the image below, I varied
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$k_a$ between 0.2 and 1. Note how the overall color of the ball increases or
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decreases in brightness when all other factors remain constant.
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{width=360px}
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\
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## Varying $k_d$
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$k_d$ is the strength of the diffuse component. It also affects an object's
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diffuse color, but at a strength that's affected by how much of it faces the
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light. Much like the dark side of the moon, the parts of the object that aren't
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pointed at the light will not receive as much of the light's influence. In the
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image below, I varied $k_d$ between 0.2 and 1. Note how the part pointed to the
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light changes the strength of the brightness as all other factors remain
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constant.
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{width=360px}
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\
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## Varying $k_s$
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$k_s$ is the specular strength. It uses the object's specular color, which is
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like its reflective component. When there is a large specular $k_s$, there's a
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shine that appears on the object with a greater intensity. In the image below, I
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varied $k_s$ between 0.2 and 1. Note how the whiteness of the light is more
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reflective in higher $k_s$ values as other factors remain constant.
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{width=360px}
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\
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## Varying $n$
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$n$ is the exponent saying how big the radius of the specular highlight should
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be. In the equation, increasing the exponent usually leads to smaller shines. In
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the image below, I varied $n$ between 2 and 100. Note how the size of the shine
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is the same intensity, but more focused but covers a smaller area as $n$
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increases.
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{width=360px}
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\
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## Multiple lights
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Multiple lights are handled by multiplying each light against an intensity
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level, and then added together. Unfortunately, this means that the intensity of
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each light can't be too bright. We rely on the image to not use lights that are
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too bright. Because this may result in color values above 1.0, the final value
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is clamped against 1.0. Below is an example of a scene with two lights; one to
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the left and one to the right:
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{width=360px}
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\
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## Shadows
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Shadows are implemented by pointing a second ray between the intersection point
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of the original view ray and each light. If the light has something obstructing
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it in the middle, the light's effect is not used.
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The soft shadow effect is realized by jittering rays across an area. In my
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implementation, a jitter radius of about 1.0 is used, and 75 rays are shot into
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uniformly sampled points within that radius. This also has the side effect that
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rays that are closer to the original ray are sampled more frequently. Each of
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these rays produces either 0 or 1 depending on if it was obstructed by the
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object. Taking the proportion of rays that hit as a coefficient for the shadow,
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we can get some soft shadow effects like this:
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{width=360px}
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\
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## Light attenuation
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Light attenuation is when more of the light is applied for objects that are
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closer to a particular light source. The function that's applied is an inverse
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quadratic formula with respect to the distance the object is from the light:
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\begin{equation}
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f_\textrm{att}(d) = \frac{1}{c_1 + c_2 d + c_3 d^2}
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\end{equation}
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Where:
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- $f_\textrm{att}$ is the attenuation factor
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- $d$ is the distance the object is from the light
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- $c_1$, $c_2$, and $c_3$ are user-supplied coefficients
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As you can see below, the effect of the light drops off with the distance from
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the light (light coming from the left):
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{width=360px}
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\
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## Depth Cueing
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Depth cueing is when the objects further from the viewer have a lower opacity to
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"fade" into the background in some sense. A good example of this can be seen in
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the image below; note how the objects are less and less bright the further they
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are away from the eye.
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{width=360px}
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\
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## Shortcomings of the model
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The Phong formula is just a model of how light works, and doesn't actually
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represent reality. There's not actually rays physically escaping our eyes and
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hitting objects; it's actually the other way around, but computing it that way
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would not be efficient since we would be factoring in a lot of rays that don't
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ever get rendered.
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Also, one needs to take care to use reasonable constants. For example, if using
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a different specular light color than the diffuse color, then it may produce
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some bizarre lighting effects that may not actually look right compare to
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reality.
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# Arbitrary Objects
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Here is an example scene with some objects that demonstrates some of the
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features of the raytracer.
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{width=360px}
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\
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