This lab is inspired by NASA's Nancy Grace Roman Space Telescope, whose objectives are “to unravel the secrets of dark energy and dark matter, search for and image exoplanets, and to explore many topics in infrared astrophysics”. A student completes a MATLAB program, in versions, to simulate the Gerchberg- Saxton (GS) algorithm, as relevant to the control of a deformable mirror in one instrument of the space telescope. NASA has produced videos on the telescope, which include a two-minute explanation of the coronagraph instrument. A related video explains a simplified instrument at the 0:18 mark.
We will explore a model of the GS algorithm and its application to a simplified instrument. In “A Tail of Two Cats” by Kevin Cowtan, the GS algorithm is explained as a way to reconstruct a missing piece of a two-dimensional (2D) image. This involves the manipulation of magnitude and phase information of the 2D Discrete Fourier Transform (DFT). For our purposes, the necessary functionality of the 2D DFT and its inverse are encapsulated in two functions, dft2 and idft2, of the coronaSimulate program.
Download and unzip V0GettingStarted.zip. In it, there are four files, including frames_v0.mat and coronagraph_v0.avi. Next, run the coronaSimulate.m file. In addition to outputting results to the Command Window and a Figure Window, it should also output a frames.mat file. The variables stored in this file, which you can load into the base workspace, should be identical to the variables stored in the frames_v0.mat file, which you can also load. Please compare the variables.
Run the coronaAnimate.m file. Assuming the Computer Vision toolbox is installed, it will output results to the Command Window and a Figure Window. A video, coronagraph.avi, is produced. Watch this video and compare it to the coronagraph_v0.avi video. They should appear identical.
Download the V1SimpleOccultation.zip file. It has frames_v1.mat and coronagraph_v1.avi result files. After unzipping, watch the coronagraph_v1.avi video. When you complete this version, requiring modifications to coronaSimulate.m only, you should be able to reproduce the results.
Complete the occultSquare, gerchbergSaxton, and getFrames functions, including the comment headers. Modify getFrames so that the axes in your results video is grayscale. Add a title to match the given result. Use “axis image off” to make the pixels square and to hide ticks and labels.
Next, modify the occultSquare function so that the central part of the image, im, is black. Implement a square shape that is width pixels wide and tall (make width the second input argument). To respect the given modular organization, do not change any function other than those specified above.
The opticalSystem function returns a second argument, which is the true phase aberration in the pupil plane of the coronagraph. A complete GS algorithm would estimate this value independently. It would start from an initial guess, at iteration zero, and approximate the true value after a maximum number of iterations. For simplicity, our simulated algorithm assumes the aberration is known.
Modify the gerchbergSaxton function so that, when it invokes idft2, it “corrects” the aberration, Dphi. Do not modify the script, especially the statement that calls the gerchbergSaxton function. When idft2 is invoked, the second argument is phase. Rewrite it so that the argument equals IMp, at iteration 0, and IMp+Dphi, at iteration maxIters. Use linear interpolation at other iterations.
Submit Version 1 of your coronaSimulate.m file, by its deadline, to demonstrate use of a version- based approach. Before submission, test it when the other files, from Version 0, are unchanged.
Download and unzip V2Occultation&Plot.zip to get frames_v2.mat and coronagraph_v2.avi files. The variables stored in the first (.mat) file, which you can load into the base workspace, ought to be identical to the variables stored in the frames.mat file once you complete this version. Modify your coronaSimulate.m file (after completing Version 1). Run coronaAnimate, which should produce a video. Compare it to the coronagraph_v2.avi video. Ideally, the videos will look the same.
Modify these functions, as well as the script: opticalSystem, occultSquare, gerchbergSaxton, and getFrames. Rename occultSquare to occultCircle and modify it so that the black central part of the image, im, is a circle having a diameter of width pixels. Use the insertShape function of the Computer Vision toolbox. Complete and edit, as needed, comment headers of local functions.
Have the occultCircle function return a second argument, mask, a logical 2D matrix whose rows and columns equal those of the uint8 2D matrix, im, a grayscale image. The mask matrix specifies occultation or simple coronagraphy. Have the opticalSystem function return it as a third argument. Entries are true for the corresponding pixels of im made black. Entries are false elsewhere.
Modify the gerchbergSaxton function to have a fourth input argument, mask, and a second output argument, errors. Next, modify the script so that the mask returned by opticalSystem is passed to gerchbergSaxton. At each iteration of the latter function, compute the sum of squared values of image, im, pixel entries (convert to double first) that correspond to mask entries that are true. Each sum is called a sum square error. Put them in a column vector, errors, of length maxIters+1.
Finally, modify the getFrames function so that the video output, coronagraph.avi, after running coronaSimulate and coronaAnimate in sequence, resembles the coronagraph_v2 video. Add a second input argument to getFrames to start. Then, in each loop iteration, plot the sum square error versus iteration number. Ideally, the image and plot should overlap on the same axes, as shown.
Use set(gca,'YDir',<expression>) to set the y-axis direction, if necessary, of an active plot axes. Use get(gca,'YDir') at a command >> or debugger K>> prompt to determine direction options.
If combining plot and image in one figure, use set(gca,'DataAspectRatio',ratio) to set the aspect ratio of pixels, where ratio is equal to [<expression1> <expression2> 1]. Determine suitable values for <expression1> and <expression2> so your video looks as expected. Each expression should be a simple ratio of an image matrix dimension and a plot axes limit.
Upon completion of Version 2, submit only your coronaSimulate.m file to eClass for marking. Ensure it works as required with the coronaAnimate.m file, which is unchanged from Version 0.
Additional Note
Until you install the Computer Vision toolbox, you can animate coronaSimulate results as follows:
>> load frames
>> movie(gcf,frames)
Once it is installed, use doc to read about the insertShape function required for Version 2.
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