CS 332 Assignment 3 Due: Monday, October 15

This assignment contains two problems on the measurement of motion in 2D images. In the first, you will complete a function to compute the perpendicular components of motion and analyze the results of computing a 2D image velocity field from these components. In the second problem, you will implement a stategy to track the moving cars in a video of an aerial view of a traffic scene, and visualize the results. The code and image files that you need are contained in the /home/cs332/download/motion folder on the CS file server. After downloading this folder, set the Current Folder in MATLAB to the motion folder.

Problem 1: Computing a Velocity Field

In class, we described an algorithm to compute 2-D velocity from the perpendicular components of motion, assuming that velocity is constant over extended regions in the image. Let (Vx,Vy) denote the 2D velocity, (uxi,uyi) denote the unit vector in the direction of the gradient (i.e. perpendicular to an edge) at the ith image location, and v⊥i denote the perpendicular component of velocity at this location. In principle, from measurements of uxi, uyi and v⊥i at two locations, we can compute Vx and Vy by solving the following two linear equations:

   Vx ux1 + Vy uy1 = v⊥1
   Vx ux2 + Vy uy2 = v⊥2

In practice, a better estimate of (Vx,Vy) can be obtained by integrating information from many locations and finding values for Vx and Vy that best fit a large number of measurements of the perpendicular components of motion. The function computeVelocity, which is already defined for you, implements this strategy. Details of this solution are described in an Appendix to this problem.

The function getMotionComps, which you will complete, computes the initial perpendicular components of motion. This function has three inputs - the first two are matrices containing the results of convolving two images with a Laplacian-of-Gaussian function. It is assumed that there are small movements between the original images. The third input to getMotionComps is a limit on the expected magnitude of the perpendicular components of motion (assume that a value larger than this limit is erroneous and should not be recorded). This function has three outputs that are matrices containing values of ux, uy and v⊥. These quantities are computed only at the locations of zero-crossings of the second input convolution. At locations that do not correspond to zero-crossings, the value 0 is stored in the output matrices. The function definition contains ??? in several places where you should insert a simple MATLAB expression to complete the code statements. See the comments for instructions on completing each statement.

The motionTest.m script file contains two examples for testing your getMotionComps function. The first example uses images of a circle translating down and to the right. The expected results displayed for this example will be shown in class. The second example, which is initially in comments, uses a collage of four images of past Red Sox players, where each subimage has a different motion, as shown by the red arrows on the image below:

   

Big Papi (upper left) is shifting down and to the right, Manny (upper right) is shifting right, Varitek and Lowell (lower right) are shifting left, and Coco Crisp (lower left) is leaping up and to the left after a fly ball. For both examples, the velocities computed by the computeVelocity function are displayed by the displayV function in the motion folder, which uses the built-in quiver function to display arrows. Your results for the Red Sox image will roughly reflect the correct velocities within the four different regions of the image, but there will be significant errors in some places. Add comments to the motionTest.m script that answer the following questions: (1) where do most of the errors in the results occur? (2) why might you expect errors in these regions? Finally, the results will vary with the size of the neighborhood used to integrate measurements of the perpendicular components of motion. (3) what are possible advantages or disadvantages of using a larger or smaller neighborhood size for the computation of image velocity?

Appendix to Problem 1: Computing the Velocity Field in Practice

It was noted earlier that in principle, we can compute Vx and Vy by solving two equations of the form shown below, but in practice, a better estimate can be obtained by integrating information from many locations and finding values for Vx and Vy that best fit a large number of measurements of the perpendicular components of motion. Because of error in the image measurements, it is not possible to find values for Vx and Vy that exactly satisfy a large number of equations of the form:

   Vx uxi + Vy uyi = v⊥i

Instead, we compute Vx and Vy that minimize the difference between the left- and right-hand sides of the above equation. In particular, we compute a velocity (Vx,Vy) that minimizes the following expression:

   ∑[Vx uxi + Vy uyi - v⊥i]2

where ∑ denotes summation over all locations i. To minimize this expression, we compute the derivative of the above sum with respect to each of the two parameters Vx and Vy, and set these derivatives to zero. This analysis yields two linear equations in the two unknowns Vx and Vy:

   a1 Vx + b1 Vy = c1          a2 Vx + b2 Vy = c2

where

   a1 = ∑uxi2    b1 = a2 = ∑uxiuyi    b2 = ∑uyi2    c1 = ∑v⊥iuxi    c2 = ∑v⊥iuyi

The solution to these equations is given below, and implemented in the computeVelocity function:

   Vx = (c1b2 - b1c2)/(a1b2 - a2b1)          Vy = (a1c2 - a2c1)/(a1b2 - a2b1)

Problem 2: Tracking Moving Objects

The motion folder contains a video file named sequence.mpg that was obtained from a static camera mounted on a building high above an intersection. The first image frame of the video is shown below:

The code file named getVideoImages.m contains a script that reads the video file into MATLAB, shows the movie in a figure window, extracts three images from the file (frames 1, 5, and 9 of the video), displays the first image using imtool, and shows a simple movie of the three extracted images, cycling back and forth five times through the images. We will go over the getVideoImages.m code file in class, which uses the concept of a structure in MATLAB, and the built-in functions VideoReader, struct, read, and movie.

Most of the visual scene is stationary, but there are a few moving cars and pedestrians, and a changing clock in the bottom right corner. Your task is to detect the moving cars and determine their movement over the three image frames stored in the variables im1, im2, and im3. (The clock has been removed from these three images.) To solve this problem, you will use a strategy that takes advantage of the fact that most of the scene is stationary, so changes in the images over time occur mainly in the vicinity of moving objects. The image regions that are likely to contain the moving cars are fairly large regions that are changing over time.

Create a new script named trackCars.m in the motion folder, to place your code to analyze and display the movement of the cars across the three images provifed. (You are welcome to define separate functions for subtasks, but this is not necessary.) Implement a solution strategy incorporating the following steps:

• For each pair of consecutive images (i.e. the pair im1 and im2, and the pair im2 and im3), find image locations where many pixels within a neighborhood around the location have a large change in brightness between the two images. Some trial-and-error exploration will be needed to determine a reasonable neighborhood size, an appropriate threshold on the brightness change, and a good value for the fraction of changing pixels in the neighborhood that is used to decide that the region may contain a moving car
• Find large connected regions of locations where brightness is changing between the two images — these regions are likely to correspond to individual cars or parts of cars (the built-in function bwlabel can be helpful here)
• Determine the approximate center of each large connected region, corresponding to the rough location of each car (the built-in function regionprops can be helpful here)
• Given the car locations identified using the image pair im1 and im2, and the car locations identified using the image pair im2 and im3, match up the locations of the cars at these two moments in time, assuming that a car region at one moment moves to the closest car region at the next moment
• Display the results of your analysis. In the solutions that I demonstrated in class, subplot was used to create one figure window with three display areas showing the first image im1 and the large connected regions obtained from analyzing the two image pairs, with superimposed red dots shown at the center of each region. A second figure window displayed im1 with superimposed red lines showing the movement of each car over time.

Hints: The file codeTips.m in the tracking folder provides simple examples of some helpful coding strategies, including examples that use the built-in bwlabel and regionprops functions, access information stored in a vector of structures, and superimpose graphics (using the built-in plot and scatter functions), on an image that is displayed in a figure window.

Be sure to comment your code so that your solution strategy is clear! (You are welcome to use a different strategy than the one outlined above.)

Submission details: Hand in a hardcopy of any code files that you wrote or modified. Drop off an electronic copy of your code files by logging into the CS file server, connecting to your motion folder, and executing the following command:

submit cs332 motion *.*