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HW4: Discovering manifolds in images: Intro to Cognitive Science SE367 2011jul-dec
Assignment 4: Discovering manifolds in images
READING:
Chapter 1 from Bishop, Pattern Recognition and Machine Learning. [covers
linear regression, entropy]
1. Isomap.
Consider the set of images given in the file randomMotion100.zip. Each image
is 800x800, and represents a position of a 2D planar robot arm for angles
theta1 in {20,30}, and theta2 in {3,29} degrees. Use the ISOMAP algorithm to
map this data into 1 to 6 dimensions. Compute the error using the
residual error function (this plot is generated by isomap.m).
- Show the graph of the residual error vs dimensions.
- Discuss the result you find.
- In the 2-D mapping, a graph is generated. Label some of the points in
this mapping with the theta1,theta2 in the textfile. Are the boundary thetas
at the boundaries of the 2D-patch?
- When we compute this graph on the images in randomMotion1K.zip, how do the
residual errors change?
- Give a table of the first 20 theta1 theta2 vs y1 y2 (Isomap)
What can you say about mapping these y to theta1 and theta2 of the arm?
2. Linear mapping and Reconstruction:
Use PCA on randomMotion100.zip to map it to 2 dimensions (keep the top two
eigenvalues). How much larger are these two eigenvalues compared to the
others?
Do the reconstruction to map a new 2D data y' back into an image x.
Consider the configurations 1 and 2 in the file randomMotion100.zip.
Find the y1 and y2 for these. Now consider y' = (y1+y2)/2,, and find
the corresponding x'. Draw it as an 800x800 image.
3. Non-Linear mapping and reconstruction:
- LLE: Use Local Linear Embedding to map the data to 2D. Now for the same
point y', obtain the original image x'. In this case, you will need to find
a set of closest neighbours yj (in 2D) and interpolate between them (express
y' = SUM wj yj). Now, in image
space, knowing the mapping xj for each yj, find x' = SUM wj xj.
- Use the same idea of local linearity to do the reconstruction in the
ISOMAP case. Draw the reconstruction x'.
- Discuss how these three images compare. What data has the system
abstracted from the input so that it is able to
reconstruct from the 2D y to the far more higher dimensional x? How is the
mapping stored in the linear case, and in the non-linear case?
4. In completeMotion2K.zip, we cconsider a full range of motion (theta1 = 0 to
135 degrees, theta2 = 0 to 180 degrees). Apply PCA and Isomap to the data,
for d=2. What can you say about the effectiveness of linear vs nonlinear
methods for this task?
FILES:
some sample code is provided for isomap etc. Codes
data files:
randomMotion100 100_angles.txt
randomMotion1K, fullMotion2K, angles
description of code files: hw4-code.html
Submit: via webpage in your-area/hw4/index.html
Due date: Friday 26 August
Submissions