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Message ID: 78     Entry time: Mon Jul 26 18:47:12 2010
Author: James K 
Type: Misc 
Category: Hartmann sensor 
Subject: Hex Grid Analysis Errors and Thermal Defocus Noise 

My previous eLog details how the noise in Hartmann Sensor defocus measurements appears to vary with ambient light. New troubleshooting analysis reveals that the rapid shifts in the noise were still related to the ambient light, sort of, but that ambient light is not the real issue. Rather, the noise was the result of some trouble with the centroiding algorithm.

The centroiding functions I have been using can be found on the SVN under /users/aidan/cit_centroid_code. When finding centroids for non-uniform intensity distributions, it is desirable to avoid simply using a single threshold level to isolate individual spots, as dimmer spots may be below this threshold and would therefore not be "seen" by the algorithm. The centroiding functions used here get around this issue by initially setting a relatively high threshold to find the centroids of the brighter spots, and then fitting a hexagonal close-packed array to these spots so as to be able to infer where the rest of the spots are located. Centroiding is then done within small boxes around each estimated centroid location (as determined by the hexagonal array). The functions "find_hex_grid.m" and "flesh_out_hex_grid.m" serve the purpose of finding this hexagonal grid. However, there appear to be bugs in these functions which compromise the ability of the functions to accurately locate spots and their centroids.

The centroiding error can be clearly seen in the following plot of calculated centroids plotted against the raw image from which they were calculated:


At the bottom of the image, it can be seen that the functions fail at estimating the location of the spots. Because of this, centroiding is actually being done on a small box surrounding each point which consists only of the background of the image. This can explain why these centroids were calculated to have much larger displacements and shifted dramatically with small changes in ambient light levels. The centroiding algorithm was being applied to the background surrounding each of these points, so it's very reasonable to believe that a non-uniform background fluctuation could cause a large shift in the calculated centroid of each of these regions.

It was determined that this error arose during the application of the hex grid by going through the centroiding functions step-by-step to narrow down where specifically the results appeared to be incorrect. The function's initial estimate for the centroids right before the application of the hex grid  is shown plotted against the original image:


The centroids in this image appear to correspond well to the location of each spot, so it does not appear that the error arises before this point in the function. However, when flesh_out_hex_grid and its subfunction find_hex_grid were called, they produced the following hexagonal grid:


It can be seen in this image that the estimated "spot locations" (the intersections of the grid) near the bottom of the image differ from the actual spot locations. The centroiding algorithm is applied to small regions around each of these intersections, which explains why the calculated "spot centroids" appear at incorrect locations.

It will be necessary to fix the hexagonal grid fitting so as to allow for accurate centroiding over non-uniform intensity distributions. However, recent experiments in measuring thermally induced defocus produce images with a fairly uniform distribution. It should therefore be possible to find the centroids of the images from these experiments to decent accuracy by simply temporarily bypassing the hexagonal-grid fitting functions. To demonstrate this, I analyzed some data from last week (experiment 72010a). Without bypassing the hex-grid functions, analysis yielded the following results:


However, when hexagonal grid fitting was bypassed, analysis yielded the following:


The level of noise in the centroid displacement vs. centroid location plot, though still not ideal, is seen to decrease by nearly two orders of magnitude. This indicates that bypassing or fixing the problems with the hexagonal grid fitting functions should enable a more accurate measurement of thermally induced defocus in future experiments.

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