# Algorithms and Data Structures

## Path tracing: sphere and triangle texture mapping

I had been planning to refine this project and offer a more detailed writeup on adding sphere and triangle texture mapping to the path tracer project, but for the time being I thought I would offer the code for download. Below are a couple of screen captures from the most recent version of the project. …

## Feature detection and tracking with an affine consistency check

The equations for detecting features, tracking them between consecutive frames, and checking for consistency using an affine transformation will be derived below using the inverse compositional approach. We will begin by deriving the equations for tracking, because this will yield some insight into which features would be good to track. The source code for this …

## kd tree construction using the surface area heuristic, stack-based traversal, and the hyperplane separation theorem

In this post we will employ the hyperplane separation theorem and the surface area heuristic for kd tree construction to improve the performance of our path tracer. Previous posts have relied simply on detecting intersections between an axis aligned bounding box and the minimum bounding box of a triangle primitive. By utilizing the hyperplane separation …

## Path tracer: thin lens, texture mapping, Fresnel equations, and smooth shading

A few new features have been added to our path tracer. The depth of field extension has been reworked slightly using the thin lens equation allowing us to specify a focal length and aperture. Fresnel equations have been added to more accurately model the behavior of light at the interface between media of different refractive …

## Path tracer with triangle primitives and binary space partitioning

UPDATE: The post below was a purely naive attempt at implementing a rudimentary bounding volume hierarchy. A much more efficient implementation using a kd tree is available in this post. We will continue with the project we left off with in this post. We will attempt to add triangles to our list of primitives. Once …

## Path tracer depth of field

This is a small extension to the previous post. We will add a depth of field simulation to our path tracer project. I ran across this algorithm at this site. Below is a render of our path tracer with the depth of field extension. Essentially, we will define the distance to the focal plane and …

## A basic path tracer with CUDA

The path tracer we will create in this project will run on CUDA-enabled GPUs. You will need to install the CUDA Toolkit available from NVIDIA. The device code for this project uses classes and must be compiled with compute capability 2.0. If you are unsure what compute capability your card has, check out this list. …

## Bidiagonalization using Householder transformations

The previous post was a discussion on employing Householder transformations to perform a QR decomposition. This post will be short. I've had this code lying around for a while now and thought I would make it available. The process of bidiagonalization using Householder transformations amounts to nothing more than alternating by left and right transformations. …

## QR decomposition using Householder transformations

It's been a while since my last post. A project I have in the works requires some matrix decompositions, so I thought this would be a good opportunity to get a post out about QR decompositions using Householder transformations. For the moment we will focus on the field of real numbers, though we can extend …

## C++ implementation of the Connected Component Labeling method using the Disjoint Set data structure

In the post before last we discussed using cvBlobsLib as a tool for blob extraction. We're going to revisit the extraction theme and look at a C++ implementation of the Connected Component Labeling method, but before we do that we're going to look at an implementation of the Disjoint Set data structure that will provide …