U.S. Agency Code: ECE 901-S1998

Status: Urgent.
 

Clearance: UWM901 - Top Secret.
 

Problem: Surveillance satellite damaged. The trajectory compensation system on the surveillance satellite (SS901) was damaged during a covert deployment mission by NASA on the space shuttle Discovery yesterday. The damage has degraded the quality of the digital images acquired by the satellite. In light of recent geo-political events, this is clearly an issue of national security. It is also the first assignment of the class. (No pressure.) Due to sensitive nature of this problem, all correspondence (verbal and written) regarding this assignment should be addressed to acting National Security Agency (NSA) director General Tull. (He is presently undercover as a professor in the ECE Department at UW-Madison. To reveal your identity and initiate discussion use the code words "the insolence of edges".) Two recent photos contain evidence of a hostile nation in possession of some of our most recent technology. Unfortunately, SS901 was only able to transmit degraded photos.

Mission: Recover original image content. NSA must confirm the content of these images before any further action can be taken. NSA is also very concerned with the nature of the methods for restoring these images. Only rigorous recovery methods will be acceptable (as discussed in General Tull's class). Preliminary experimentation and background discussion is necessary to justify your approach.

 
Intelligence: Only a noisy (motion) degraded observed image is available. The satellite transmits images at different resolutions as part of its compression scheme. Several similarly degraded images at different resolutions are available. The noise characteristics are constant for each resolution. The distortion is symmetric and anti-casual. New intelligence will be disseminated by General Tull on a need to know basis along with mission updates.

Operation Hamlet is due March 3, 1998. Reports received after 5pm will be disavowed. (After all, national security is at stake.) The following pages describe the research activities necessary to convince NSA of the restored image content. Good luck.
 
 

• Estimate H from distorted image g.

 

• From f_A (512x512) estimate the eigenvalues of H from the DFT domain of g. (Hints: Distortion magnitude is evident in a line of the image in the magnitude DFT domain. The PSF is symmetric and due to motion blur.)
• Plot the first M eigenvalues of H for the three cases (best, closest and worst) estimate of H.
• Repeat for each resolution in group A. (Hint: The extent of the motion blur estimate at each resolution should differ by a factor of two.)
• Explain in detail the difficulties and differences in estimating H at different resolutions.
• Show why this is estimation procedure is more difficult for images in Group C.
 
• Form generalized inverse H⁺(T_k)

 
• Using your best estimate of H at each resolution, take the reciprocal of its eigenvalues, l_i, for |l_i| > T_k, where T_k = _{10^-k (k = 1,3,5,7) and zero otherwise.}
• _{Plot first M eigenvalues of H⁺(Tk) for each k at each resolution. (For NxM image, M is number of columns in matrix)}
 
• Restore image using generalized inverse.
• Apply generalized inverse H+(Tk) for each group (A,B,C) of images at the 512x512 and 256x256 resolution. Impose positivity. Print resulting images.
• Explain the observed phenomenon.
 
• Form the regularized inverse H^~(ak)
• H^~(ak)=(H^TH + akC^TC)^-1 H^T - noting that the addition and multiplication of block circulants result in a block circulant matrix, plot the first M eigenvalues of the regularized inverse for ak = 10-k, k = 1, 3, 5, 7.
• Compare and contrast the eigenvalue structure of the inverse operators H^~(ak) and H⁺(Tk) at the 512x512 resolution when C is the 2D Laplacian and when C = I-H (I-H, why?)
• Using the method prescribed by Hunt, obtain an estimate of the "optimal" regularization parameter for each resolution. Is the regularization parameter a function of the resolution? Why (not)?
 
 
• Restore image using regularized inverse.
• Apply regularized inverse H^~(a_k) for each group (A,B,C) of images at 512x512 and 256x256 resolution for each a_k. Impose positivity.
• _{Print and label restored images. Note regularization parameter, resolution, etc.}
 

 

• Implement regularized iterative restoration [Hunt/Kats.]
• Establish convergence criteria for iteration and report (optimal) relaxation parameters for each experiment.
• Use Hunt approach to determine regularization parameter. Additional intelligence will be made available regarding the selection of the regularization parameter.
• Apply to each group (A,B,C) of images at 512x512 and 256x256 resolution for each a_k. Impose positivity at each iteration.
   
• Report. (typed, Latex, Word, etc.)
• Include all plots and results (600dpi) or better.
• If in prose (preferred presentation), highlight paragraphs where issues and answers to questions are addressed (margin notes are good).

1. Image data: in PGM format. This is a universal translation format for raw images (a.k.a. NetPBM or PGMPLUS). Of the many flavors, we only consider the raw ascii storage version. See enclosed specification. In this case, each pixel is stored as an 8 bit (unsigned char). This is the standard format for image processing research. M-files entitled "readpgm.m" and "savepgm" will be provided for Matlab users to read a pgm file into a Matrix and save a matrix as a pgm file, respectively.

 

2. Viewing images: Unix - use XV, Matlab -- use image(f). Type "help image" in matlab to learn more about the range of values for f. DO NOT USE imagesc(f) for viewing restored images since it distorts (scales) the histogram of the image. However, imagesc(f) is suggested for viewing difference images.

 

 

3. Available images: The images g¹ and g² are in two groups,