Project 4: Antialias (Algo)

You can find the handout for this project here. Please skim the handout before your Algo Section!

You may look through the questions below before your section, but we will work through each question together, so make sure to attend section and participate to get full credit!

Please read the project handout before going to algo section and working on these questions!

Discrete 2D Convolution

Suppose that you are given a 1D image vector , representing a 2D image of height and width . You are also given a 1D kernel vector , with radius . You may ignore edges for now.

Write out pseudocode to slide the kernel over the 2D image. At each kernel position, calculate the index of each pixel in the 1D image vector () that the kernel () overlaps with.

Kernel positioned horizontally over a 2D image.
Figure 2: An example of a kernel positioned horizontally over a 2D image.

How would you need to adjust this pseudocode if the kernel were positioned vertically instead?

Duality of Domains

Visualizing Sinc and Box Duals

Take a look at the box function in figure 5 (left) below. Its dual in the frequency domain is the sinc function, in figure 5 (right). Note that there are no labels on the axes, by design.

A box functionA sinc function
Figure 5: A box function with its dual, a sinc function.

Draw out a sketch of the dual of the box function in figure 6. Your drawing doesn't need to be perfectly accurate; but enough to show that you understand the concept.

Another box function, about twice as wide as before
Figure 6: Another box function, about twice as wide as before.

When you are finished, explain your group's thought process to the TA.

Approximating Sinc

What do we usually use to approximate the sinc function, and why do we have to make this approximation when translating these theoretical concepts into code?

Discuss with your group and then call over the TA to check your understanding.

Sampling in the Frequency Domain

If we're sampling a continuous function at a frequency of samples per unit, what is the largest frequency we can represent, according to the Nyquist limit?

Frequency Plots

Now, examine the following frequency plot of a 1D, continuous signal. If you were going to sample this signal at a rate of 8 samples per unit, to avoid aliasing, you would use what we know about the Nyquist limit to pre-filter it.

Draw out the new frequency plot after someone has performed this pre-filtering step optimally. Make sure to include any relevant amplitude and/or frequency values.

A frequency spectrum. It plots amplitude against frequency.
Figure 7: A frequency spectrum. It plots amplitude against frequency.

Ray Differentials for Spheres

Review: Texture Coordinates

Recall that we can express coordinates on the surface of a sphere using spherical coordinates and . For a sphere, what is the relation between texture -coordinates and these spherical coordinates and ? (Hint: These lecture slides on the surface parameterization of a sphere might be helpful.)

From Texture to Object Space

Using the equations that we found in 3.1 and the definition of and given in the slides, express our sphere's texture coordinates in terms of its object space coordinates .

Calculating Differentials

Calculate and : that is, the differentials of our sphere's texture coordinates with respect to its object space coordinates . (You'll need to calculate six partial derivatives in total.)

The following derivatives might be helpful:

Bilinear Sampling

bilinear sampling
Figure 8: Image coordinates vs texture (UV) coordinates.

Suppose we have texture of size , which is repeated times horizontally and times vertically. Write out the formula for bilinear sampling of the texture given UV coordinates .

Submission

Algo Sections are graded on attendance and participation, so make sure the TAs know you're there!

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