Problem 1000. Image Processing 004: Zernike Coefficients

Created by Richard Zapor

Solve Later

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Pages 13 and 14 give the term definitions implemented in the zern_data_v01.m routine provided in the function template.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"-1\"/><w:attr w:name=\"width\" w:val=\"-1\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>ANSI Standard Zernike Chart:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Algorithm Description:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[M=surface array to be fit by Zernike terms; \n-valid data is circle of normalized radius<=1\nUse the R and T arrays to create the A array\nR=radial normalized distance from center of array\nT=angular radian value from center of array\nA=fitting array [valid_data_pts,terms]\n\nCreate A * z_fit = M(valid_data_pts)\n[valid_pts,terms] * [terms,1] = [valid_pts,1]\n\nLoop term=0:terms-1\n- Create the 2-D array of z-terms uisng z=zern_data_v01(term,R,T)\n- A(:,term+1) = z(valid_data_pts)\nEnd Loop\n\nSolve for z_fit: pinv(A)*M(valid_pts) or A\\M(valid_pts)]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Inputs:</w:t></w:r><w:r><w:t> [M N R T] where N is number of Zernike terms for fit</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Outputs:</w:t></w:r><w:r><w:t> [v] Zernike fit vector of length N</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Creation of the M surface and expected v fit vector can be seen in the Test Suite.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" 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"}]}

Fit Zernike Terms to a wavefront.The class notes of <a href="http://www.visualopticslab.com/OPTI535/Lectures/Class06_08.pdf">Schwiegerling UofA</a> on ANSI Standard Zernike terms is the basis for this challenge. Pages 13 and 14 give the term definitions implemented in the zern_data_v01.m routine provided in the function template.<img src="https://sites.google.com/site/razapor/_/rsrc/1350774824076/matlab_cody/Zernike_UA_480.jpg">ANSI Standard Zernike Chart:Algorithm Description:<pre class="language-matlab">M=surface array to be fit by Zernike terms; 
-valid data is circle of normalized radius&lt;=1
Use the R and T arrays to create the A array
R=radial normalized distance from center of array
T=angular radian value from center of array
A=fitting array [valid_data_pts,terms]
</pre><pre class="language-matlab">Create A * z_fit = M(valid_data_pts)
[valid_pts,terms] * [terms,1] = [valid_pts,1]
</pre><pre class="language-matlab">Loop term=0:terms-1
- Create the 2-D array of z-terms uisng z=zern_data_v01(term,R,T)
- A(:,term+1) = z(valid_data_pts)
End Loop
</pre><pre class="language-matlab">Solve for z_fit: pinv(A)*M(valid_pts) or A\M(valid_pts)
</pre>Inputs: [M N R T] where N is number of Zernike terms for fitOutputs: [v] Zernike fit vector of length NCreation of the M surface and expected v fit vector can be seen in the Test Suite.

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Rafael S.T. Vieira on 12 Sep 2020 at 0:41

Could you please fix the function urlwrite? Or remove it...adding the code for the function zern_data_v01(j,r,t) at the end of the live script. Good problem.

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Problem 1000. Image Processing 004: Zernike Coefficients

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Problem 1000. Image Processing 004: Zernike Coefficients

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