CS 184: Computer Graphics and Imaging, Spring 2019

Project 1: Rasterizer

Bryan Tong, 26548098


For this project, I implemented a rasterizer that can render images with various forms of sampling and antialiasing. The rasterizer can even transform, rotate, scale objects, and work with both svg's with texture maps and png's alone. It features a 3-line test triangle rasterizer, a barycentric coordinate converter, uv texture mapping, and pixel and level sampling. We implemented pixel and level sampling up to the degree of featuring trilinear filtering.

This project was really neat to me - we went from stone-cold C++ code to implementing a triangle rasterizer from mathematical concepts in class, to antialiasing, performing sampling and interpolation - that's crazy. Implementing something like trilinear filtering, which I have seen extensively in video games, help tied the theoretical concepts of computer graphics to something concrete.

Section I: Rasterization

Part 1: Rasterizing single-color triangles

Rasterizing triangles is a key basic step to computer graphics, involving the processing of primitive points on an integer grid representing pixels. To rasterize triangles, we performed a test to determine if one of these points (iterating through many, eventually) was inside or outside the triangle via a 3-line test. We did this by building on-top of the provided line function, processing 3 points at a time: p0, p1, p2, and breaking them down into their respective (x,y) coordinates and checking the edges of a triangle. In my project, I did this by abstracting this process with helper functions: I made a helper function to determine the polarity of a line being either positive or negative, and then I applied this to all three edges (as a result of having three points). I then did a simple test to check if they are all positive or all negative, which indicates that a point must be inside the triangle; else, the point may lie on a plane that is intersected by 1 or 2 edges, but not all 3.

If there is a triangle, we feed in the point coordinates of points that are inside the bounding box of the triangle. It is not possible for this to be worse in runtime or less efficient than an algorithm that checks every sample within the bounding box entirely.

I did a minor optimization (maybe not enough for extra credit), but I swapped the ordering of the x and y pixel loops such that y is processed first. I noticed that we performed operations such as matrix[y][x] = ... for checks, so I ordered it such that y gets evaluated first and minorly avoids redundant operations.

test4.svg, showing a disconnect between pixels on default settings

Part 2: Antialiasing triangles

To implement supersampling for AA, I divided each pixel into its subsamples, via mathematical manipulation of 1 / float((sqrt(sample_rate) * sqrt(sample_rate)) to determine an appropriate amount of subsamples. I then similarly computed a subshifter that helped me identify each subpixel's center. I performed the inside-triangle test on these subsamples, summed their RGB values, and then divided that by the total number of subsamples per sample to return our RGB value of the pixel.

By doing these operations, we are able to create a primitive form of antialiasing where the jagged edges become smoother and breaks / gaps, such as in our pink triangle, also become more filled in and pleasing to the eye.

To implement this, I had to heavily modify rasterize_triangle, which actually helped me catch a bounds bug in my triangle inequality method. I also had to implement the color summing code to help fill in fragments in the rasterization pipeline.

You can visually see supersampling's effects in the images below. We apply supersampling to weight subsamples of pixels to create colors that did not exist in the original image, yet are crucial to a natural viewer to perceive images as smooth and without jaggies. In the example of the pink triangle, we can see where the supersampled pink elements on the jaggies appear to be a far lighter pink, an effect of the supersampling.

Sample rate of 1, we see a strong disconnect.
Sample rate of 4, a middle line of pixels appears.
Sample rate of 16, nearly filled in via antialiasing.

Part 3: Transforms

The inspiration of this dancing cubeman is music. I was sitting in Blue Bottle Coffee, listening to some groovy beats. Headphones on blast, I tuned out the world and tweaked my groovy dancing cubeman to best represent my inner-self. It also used a nice combination of every transformation we implemented.

A dancing cubeman

Section II: Sampling

Part 4: Barycentric coordinates

Barycentric coordinates allow you to interpolate a color via the weighted distance from three points, traditionally defined as alpha, beta, and gamma, representing red, green, blue respectively. They are also interestingly affine coordinates. The weighting occurs as you travel away or towards a coordinate, the color value respectively increases or decreases in ratio. This is due to a barycentric coordinate needing to sum to 1, e.g.: alpha + beta + gamma = 1, so the other two respective colors fill in the missing amount. Thus, the result we see is a smoothly blended color triangle.

Diagram of Barycentric Coordinates from CMU

This is the color wheel that my rasterizer program created.

A color gradient wheel

Part 5: "Pixel sampling" for texture mapping

Pixel sampling uses barycentric coordinates to apply a texture map onto an image: a .png in our scenario.

We converted our (x, y) pair points to barycentric coordinates, and then converted everything to uv coordinates in the end. For the two pixel sampling methods we implemented, nearest neighbors was fairly simple.

In nearest neighbors, you select the closest texel and return the color.

Meanwhile, for bilinear sampling, you take the 4 nearest samples and relationalize them as pixel coordinates. Then, I chose to split their centers via fractional coefficients to obtain the location of the center of the 4 nearest samples, plus their fractional values to use as weighting. We weighted the 4 colors of the nearest texels, performed some arithmetic, and then returned a color based off of these calculations.

There is quite a large difference, both visually and computationally. Bilinear takes at least 4x more work than the naive NN method, due to grabbing the 4 nearest samples, plus additional computational time depending on methods used to perform the weighting calcultions. Visaully, we can see even before antialiasing is applied that bilinear sampling looks quite good and smooth. The white dots are already smoothed into a continuous line, and has done quite a good job with making the pixel-inspector's image more legible to the untrained eye. At even 1 sample per pixel, bilinear sampling (in my eyes) outperforms NN @ 16 samples per pixel. Bilinear sampling @ 16 samples per pixel looks extremely smooth, but also sharper as a whole.

Nearest sampling at 1 sample per pixel
Nearest sampling at 16 samples per pixel
Bilinear sampling at 1 sample per pixel
Bilinear sampling at 16 samples per pixel

Part 6: "Level sampling" with mipmaps for texture mapping

Level sampling is when we use a specific sample from a series of images, each level with resolutions respectively lowered by the power of 2. Level sampling helps us pick a high-level mipmap sample for near objects, and a low-resolution, deeper depth sample for minified textures in the distance. In laymans terms, it helps us easily and efficiently apply a varying level of details an image, at the cost of increased memory usage. It also helps us alias scaled-down images without completely blurring every detail away.

We implemented it by first calculating changes in our (x, y) coordinates, which again underwent barycentric and uv conversion. Then, we found the differences in our du and dv vectors, scaled it by the proper level's width and height, and finally plugged it into the equation given in lecture to output a number. I rounded this number and used it to perform operations respective to nearest level or bilinear levels.

As for efficiency, level sampling is known to take additional memory (theorhetically 33%), but the FPS and CPU impact was not as high as I expected. I suspect the large lack of memory increase too in our examples is due to how simple our rasterizing program is on a modern computer. (Run on my 2015 Macbook Pro, with a AMD Radeon card.) Level sampling had barely any load impact compared to pixel sampling, but the only combination of modes that made marked difference in loading time was for fun when I played around with trilinear filtering. However, pixel sampling (especially combined to become trilinear) is clearly more effective at AA.

Comparison time!

  • L_ZERO and P_NEAREST: Averaging 59 FPS, 86% CPU, 340 MB.
  • L_NEAREST and P_NEAREST: Averaging 50 FPS, 87% CPU, 343 MB.
  • L_ZERO and P_LINEAR: Averaging 55 FPS, 85% CPU, 343 MB.
  • L_NEAREST and P_LINEAR: Averaging 47 FPS, 87% CPU, 343 MB.
  • TRILINEAR: Averaging 39 FPS, 88% CPU, 343 MB.

    Please look at the beauty of trilinear filtering

    Section III: Art Competition

    I didn't create any art, but I'll be sure to vote! :)