Experimentation of Centroid Stability and Simulated Noise, statics and Dynamics

1. Experimentation of Centroid Stability and Simulated Noise, statics and Dynamics Kevin Kelly Mentor: Peter Revesz

2. Brief Introduction • Importance of Project: Beam stability is crucial in CHESS, down to micron-level precision • The beam position is measured using a video imaging system by determining the intensity centroid. • We measured how parameters like the image frame averaging number and other acquisition settings affect the stability of the measured centroid, modeling the X-ray luminescence with an LED light source on an optical bench. • We analyzed data measured with USBChameleon to calculate the sigma of the centroid position, lower sigma relating to higher precision. • Noticed a trend – the greater the frame average, the lower the sigma.

3. Chameleon GUI Inputs ROI Profile of Image Outputs Centroid Position Trace

4. Experimental Setup • We have carried out our experiments using the model light source (LED) mounted on a linear precision slide on an optical bench for maximum mechanical stability • To reduce any potential outside effects on the experiment (airflow, external light, etc.), we added a special shielding cover. • Using the USBChameleon program we obtained statistical data about pixel intensities and centroid position under various experimental conditions.

5. CCD Camera, Mounted to the Optics Bench Chameleon by Point Grey Research, pixel size ~3.5 mm Resolution: 640x480 (low), or, 1280x960 (high) The LED Light source, mounted and stabilized Experimental setup, covered in metal shield to reduce any airflow from affecting the light source

6. Parameters to Test Our next parameter to test was Shutter Time, or the length of exposure for each image. Because longer exposure time means more light incident on the CCD, the higher shutter time, the brighter the image. • Tested 60 combinations of Frame Average and Shutter Time: • Frame Average: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 • Shutter Time: 25ms, 40ms, 55ms, 70ms, 85ms, 100ms We repeated this for Low Camera Resolution and High Camera Resolution to have data for both settings as well. • Other parameters not adjusted include: • Gain • Camera Lens • Light source symmetry • Aperture

7. The instability of the centroid comes from a variety of sources: • Experimental Conditions: Mechanical instability of the light source, vibrations, airflow in experimental tunnel. • Errors in CCD: Read Noise, Photon Noise, Dark Noise • Experimental Paramters: Adjusting Frame Average, Shutter, Type of centroid calculation, etc. 2s A typical plot of the centroid position over time, static LED position. After fitting a trend, we analyze the residual and calculate a standard deviation from that (s). Instead of trying to separate and individually analyze the sources of noise in experiments, we opted to just measure the standard deviation experimentally and attempt to look at the noise sources through different means.

8. Results • Observed the same trend for Shutter Time as for Frame Average – inverse relationship. After analyzing, the trend was noticed to be an inverse-square-root. The equation is this: Where F is the Frame Average Value and S is the Shutter Time, in milliseconds. Some key measurements we needed to keep track of: Shutter Time, Frame Average, Sigma (obviously), but also the ROI Size and the average FWHM of the profile throughout each trial. These will be important later. For static LED the best precision we measured ~0.07 mm !

9. Analysis of Pixel NoiseStatic Image • Added a new functionality to USBChameleon, PixelSave. • Used this on a slice of the image for all the combinations of frame average and shutter time mentioned above. • The results were then analyzed to see a trend between a given pixel’s average intensity and intensity distribution. • We repeated this measurement a number of times (typically 200) to obtain statistics ROI of Pixel Intensities being saved Pixel Intensities across ROI

10. Typical plot of Standard Deviation of Pixel Intensity v. Average Pixel Intensity with trendline. • Two important things to note: • Each pixel’s histogram resembled a Gaussian distribution. • Increasing Average Intensity leads to increasing Standard Deviation. – close to a square root dependence. • We used the measured spix-intvs. intensity values in the Monte Carlo Simulations Histograms of single-pixel statistics, one at low intensity, one at high.

11. Moving from Static to Dynamic • Aliasing effect 1: The pixel intensity digitalization, because it is to 12-bit accuracy, introduces “rounding.” • Aliasing effect 2: Averaging the pixel intensity over a finite pixel size results in a “jagged” image profile • As a result, the image is slightly distorted, and the distortion changes as the image moves. • The overall result of this is a (periodic) artifact in the centroid position during image motion. • To verify this, we used the motor-controlled slide that the LED is mounted on to perform our experiments, taking steps of 5, 10, and 15 microns, as well as a steady state to compare.

12. Aliasing Pictures Beginning Halfway End

13. Just looking at the trace of the measured centroid, the position seems perfect! But when we take a closer look at the residual, we observe that there are two frequencies of oscillation. Trace of Centroid, moving 10 micron steps The easiest way to analyze this residual is an FFT, Fast Fourier Transform. The result of this is a plot of frequency v. magnitude. The magnitude is related to the amplitude of the oscillation at that given frequency. Residual of Above Graph

14. FFTs LED moving by10 micron steps, Low-Res, Normal Centroid Control – Steady, unmoving, Normal Centroid LED moving by10 micron steps, Low-Res, Squared Centroid LED moving by15 micron steps, High-Res, Normal Centroid

15. Methods to Reduce Aliasing “Jigsaw” Enlarge 45-Degree Tilt Average

16. Simulation • Developed a Monte Carlo simulation program to create simulated 2D profiles with randomized “noise” based upon measured data and calculate the centroid, similar to what USBChameleon does. • For this I used the tables created from the single-pixel intensity statistic measurements. Ideal Gaussian Noise added for Randomization

17. Simulation GUI • After all of the code was written, I wrote a GUI for it for appearance’s sake and ease of use. • It has all of the inputs necessary and displays the simulated image over time, with the 12-bit greyscale converted to RGB (for visualization purposes). The code also enables the user to move the Gaussian Profile with added noise as the centroids are calculated taking into account the discrete levels of the grayscale averaging over individual pixels.

18. Simulation ResultsStatic Image Simulating the low camera resolution experiment, the results had the same trend as the experimental values, just with a lower value, roughly a quarter of the measured sigma. The equation was this:

19. Simulation ResultsDynamic Image FFT of Ideally Simulated Centroid moving 10mm steps over 8mm, Low Resolution FFT of Ideally Simulated Centroid moving 10mm steps over 8mm, High Resolution

20. Conclusion • Frame Average and Shutter both have a significant effect on the stability of the centroid, at the tradeoff of measurement time. • The noise of the pixel intensity closely follows a square root dependence on the intensity, as expected. • Analyzing the dynamic (moving) light source, we saw effects of aliasing in the centroid position. We believe that this is a combined effect from finite pixel sizes and the digitalization of the pixel intensity. • To characterize the aliasing effect of the centroid position, we utilized FFT techniques. This FFT analysis revealed oscillations related to the pixel size and oscillation related to the imperfection of the lead screw of the motor-controlled slide. • Monte Carlo simulation of the 2-Dimensional image profile with added noise (using measured values) resulted in similar trends of centroid stability as the experimental data for both static and dynamic images.

21. Questions?