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# CSCI 4830/5722 Assignment 4 Dynamic Programming

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Task 1 (20 points): Implement the Longest Common Subsequence (LCS)
Given two sequences, X ={x1; … ; xm} and Y ={y1; … ; yn}. Find a subsequence common
to both whose length is longest. A subsequence doesn‘t have to be consecutive, but it
has to be in order.
The LCS problem has 2 versions:
• The Simple version, requesting only to find out the length of the longest
common subsequence.
• The Complete version, requesting to find out the sequence itself.
For this assignment, you need to implement the Complete version using the Dynamic
Programming approach.
X = {x1, … xm}
Y = {y1, …,yn}
Xi = the prefix subsequence {x1, … xi}
Yi = the prefix subsequence {y1, … yi}
Z ={z1, … zk} is the LCS of X and Y .
LCS(i,j) = LCS of Xi and Yj
We can recursively define the LCS as:
Use the recursive definition to set up your memorization table and compute the size
of the longest subsequence and the characters that are part of it.
LCS(i, j) =
0 if i = 0 or j = 0,
LCS(i −1, j −1)+1 if i, j > 0 and xi = yj
,
max(LCS(i −1, j), LCS(i, j −1)) if i, j > 0 and xi ≠ yj
.
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H O R S E B A C K
S N O W F L A K E LCS = OAK
Task 2 (40 points): Calculate stereo disparity using the DP (as outlined below)
Here, you will implement a stereo algorithm that uses dynamic programming. This
algorithm enforces the ordering constraint, the uniqueness constraint, and matches
individual pixels based on a cost function. Every pixel in one image can either match
one pixel in another image, or be marked as occluded.
Note: this algorithm assumes the image intensities in the left and right image fall in
the range 0 to 1.
Part A (20 points): Disparity matching along each epipolar lines
Just like in Assignment 3, you will use the files provided with the script
DepthEstimationFromStereoVideoExample. Your DP algorithm will
replace the disparity.m built-in Matlab function. For each image pair, you seek to
output a disparity map for the left image indicating the disparity to the right image.
The cost function is defined such that the penalty for matching two pixels is the
square of the difference in their intensity. The penalty for a pixel being occluded is a
fixed value, occ.
The images (frames in the video) are already rectified, so that the epipolar lines are
horizontal scan lines of the input images. Thus, you just need to run the DP stereo
algorithm on each corresponding line/row in the two images. You will need to call
your function once for each epipolar line. Your function should have the form:
D = stereoDP(e1, e2, maxDisp, occ)
The recommended value for occ is 0.01. For maxDisp, you can start with the
value 64 (this is the maximum disparity value as resulting from using the built-in
Matlab disparity.m function). Feel free to try a lower value and notice if your
disparity maps improves.
Algorithm:
Consider two scanlines Il(i) and Ir(j), 1 ≤ i, j ≤ N, where N is the number of pixels in
each line (the process will be repeated for each row of the image). Pixels in each
scanline may be matched or skipped if they are considered to be occluded, in either
the left or right image).
Let dij be the cost associated with matching pixel Il(i) with pixel Ir(j). Here we
consider a squared error measure between pixels given by:
dij = (Il(i) − Ir(j))2
The cost of skipping a pixel (in either scanline) is given by a constant occ.
We can compute the optimal (minimal cost) alignment of two scanlines recursively
as follows:
1. D(0,j) = j * occ
2. D(i,0) = i * occ
3. D(1, 1) = d11,
4. D(i, j) =
= min(D(i−1,j−1) + dij , D(i−1,j) + occ, D(i,j−1) + occ)
Note: just like in the LCS complete problem, you will need to save which “direction”
was used for the calculation of each D(i, j)value: North, West, or Northwest.
Part B (10 points): Backtracking
The goal is to find the optimal alignment (and thus the disparity) by backtracking.
Starting at (i, j) = (N, N), trace your path of minimum cost all the way to (1,1). You
will need to use the “directions” saved in part A. Selecting (i − 1, j) corresponds to
skipping a pixel in Il (a unit increase in disparity), while selecting (i, j − 1)
corresponds to skipping a pixel in Ir (a unit decrease in disparity). Selecting (i − 1, j −
1) matches pixels (i, j), and therefore leaves disparity unchanged.
Part C (10 points): Displaying the disparity map
For display purposes, the disparity values should be normalized and mapped to the
range [0,1] and, to distinguish valid disparities from occlusions, you should colorize
the image so that occluded pixels are shown in color while the rest of the disparity
map is shown in grayscale. Here is pseudo-code for scaling the values appropriately
and showing occlusions in a different color:
function [d_color] = display_dmap(d)
% 1. Map disparity into the range [0,1]
% max_d = the maximum calculated value of disparity;
% min_d = the minimum calculated value of disparity;
% scale the disparity values by subtracting the minimum
% value min_d and dividing by the difference beween max_d
% and min_d
% 2. Colorize occluded pixels to be red
% dColor = color image where each RGB layer is equal to the
% scaled disparity matrix (values between 0 and 1)
% find the positions/indices where each of the 3 values of
% dColor is equal to NaN, and store them in a variable
% replace the values of these positions with:
% dColor(at position in R layer) = 1;
% dColor(at position in G layer) = 0;
% dColor(at position in B layer) = 0;
% 3. Display dColor image using imshow
Part C (extra credit 20 points): Different Cost Metrics
Up to this point, we have used the squared difference in pixel value as our matching
cost. In this section we will extend our definition of matching cost to include
neighborhoods of pixels. We will evaluate two different cost metrics:
1. (10 points) Sum of Squared Differences (SSD) and
2. (10 points) Normalized Cross Correlation (NCC).
We wish to compute the SSD/NCC match cost between pixel (x1, y) in image 1 and
pixel (x2, y) in image 2. We first extract a window W1 centered at pixel (x1, y) from
image 1. We then extract a window W2 centered at pixel (x2, y) from image 2. The
size of the two windows should be the same. Use the same definition of window
matching with SSD and NCC as in Assignment 3.
For this assignment (extra credit), evaluate your algorithm using the SSD and NCC
match costs with window sizes of 3×3 and 5×5 (four disparity maps total, 5 points
each). You can start with the same occlusion penalty (occ) as for 1-pixel SSD, and
then try to change this value and see how it affects your disparity maps.
Since evaluating these cost functions can be computationally intensive, you may find
it helpful to optimize your implementation to get acceptable run-times. The
following list of suggestions may be of help:
• Adjust the values for the minimum and maximum disparity for each epipolar line
and only evaluate the match cost for pixels in this disparity range. You can use the
values obtained at the first pass and then reduce the search range accordingly.
• Think of ways to compute any part of the match cost using vectorized code rather
than loops.
• Take advantage of Matlab’s profiling tool (available under Desktop → Profiler in
the menu). This will isolate the slowest parts of your code.
Implementing the first suggestion is probably sufficient to make your algorithm run
in a reasonable amount of time.
Task 3 (20 points): Calculate stereo disparity using the DP (cones images)
Task 4 (20 points): Calculate stereo disparity using the DP (teddy images)
The left and right images for the cones and the teddy sets, as well as the left and right
disparity maps (ground truth) can be found here:
http://vision.middlebury.edu/stereo/data/scenes2003/
Submitting the assignment:
Make sure each script or function file is well commented and it includes a block
comment with your name, course number, assignment number and instructor name.
Zip all the .m files together and submit the resulting .zip file through Moodle as
Assignment 4 by Wednesday, November 2nd, by 11:55pm.