ER-75, WFCam Guider Centroid Algorithm

Introduction

The contributors to the background noise level are DC and rms dark current, amplifier read noise and DC and rms sky brightness. The digitizer setup also offsets the digitized data by a fixed amount across the entire frame. Dark current, digitizer offset and amplifier read noise are static parameters (assuming that the temperature of the CCD is maintained constant) and can be measured. Sky brightness is dependent primarily on the phase of the moon, the location of the moon in the sky and the sky glow from nearby population centers. These contributors vary not only from night to night but can even vary over the course of a night, perhaps even over the period of a long exposure. Therefore in order to implement the centroiding algorithm set out in ER-60 it is necessary to have a dynamic method for determining the threshold level. This report presents an algorithm for tracking the centroid of a star incorporating a threshold algorithm which can be used to dynamically set a threshold value for each acquired image.

Threshold Method

Star Image Before and After Thresholding.
Figure 1.

Conceptually it is easy to describe a method for setting the threshold. By starting at highest point on the image and walking down the "mountain" the level of the noise is reached when the steep descent stops and levels out. This point can be found mathematically using a second moment computation. By starting at the highest point in the image a threshold is set at one A/D level below the peak and the second moment of the coordinates of all points above this level is computed. The threshold level is then reduced by one A/D level and the second moment is recomputed. The result of this computation on the data in Figure 1 is shown in Figure 2. Notice that as the threshold decreases from its peak of 200 the second moment stays almost at zero until around 30 where it takes a sharp turn upward. This is the level of the background noise and the proper level for the threshold setting (the model noise value for this data is ).

Second Moment vs. Threshold Level
Figure 2.

Sensing the location of the knee of the curve can be done by monitoring when the slope of the curve crosses a threshold. It is necessary to filter the second moment data to make it smoother to minimize the noies in the first derivative. Figure 3 shows a filtered second moment and the first derivative of the filtered data over a smaller range. Notice that by simply watching for the first derivative data to cross a preset level (100 was chosen) a value for the threshold can be found.

Filtered Second Moment and First Derivative
Figure 3.

Centroid Algorithm

1. Point the telescope such that an appropriate guide star is in the field of view of the guide camera and start tracking. This star should be the brightest star in the field.
2. Set the exposure time for the guide camera. Ideally this exposure time should create a peak intensity greater than 50% of camera saturation (but never saturated). This may not always be possible on dim stars and the actual peak intensity might be much lower.
3. Allow the guide camera to integrate and acquire the first guide frame.
4. Search the entire guide camera image and find the brightest pixel.
5. Set a 15x15 pixel window around the brightest pixel.
6. Threshold the data
   1. Find the values of the lowest and highest data points in the window. Set these value to Level_Low and Level_High respectively.
   2. Search the window for the points with the value Level_High.
   3. Compute the sum of the squares of the distances of these points from the center of the window put this value in Current_Sum.
   4. If this is the first point store Current_Sum in Second_Moment(Level) else store Current_Sum+Second_Moment(Level-1) in Second_Moment(Level).
   5. Level = Level - 1.
   6. Go to 6.2 until Level = Level_Low.
   7. Filter Second_Moment using a normalized moving 5 point center weighted filter (weights of 1,2,3,2,1, Normalization factor = 9), put the result in Filtered_Second_Moment.
   8. Set Level = 2.
   9. Compute Filtered_Second_Moment(Level) - Filtered_Second_Moment(Level - 1) beginning at Level_Low until the result is less than 100.
   10. Set Threshold = Level.
   11. Set every point in the Window with a value less than or equal to Threshold to zero, set all other points to Value - Threshold.
7. Locate the Centroid
   1. Compute the intensity weighted first moment for the x direction. This is done by summing the product of the intensity of a pixel times it's x-distance from the center of the window and dividing this product by the sum of all of the intensities in the window. Save this value as x_centroid.
   2. Compute the intensity weighted first moment for the y direction. This is done by summing the product of the intensity of a pixel times it's y-distance from the center of the window and dividing this product by the sum of all of the intensities in the window. Save this value as y_centroid.
   3. Scale x_centroid and y_centroid to real units by dividing by the telescope plate scale factor.
8. Store the centroid position, this will be the reference position for the centroid.
9. Set the guide camera controller to acquire the same 15x15 window identified in step 5.
10. Integrate
11. Read out the subarray.
12. Begin next integration.
13. Threshold the data
14. Locate the centroid
15. Compute the distance and direction from the computed centroid to the reference centroid.
16. Send the offset distance and direction information to the TCC.
17. Go to 11.

NOTE: Because the window used to track the centroid is so small, it is not necessary to flat field the camera. This assumes that the camera response is relatively uniform across the centroid window which should be a safe assumption. If camera testing determines that hte camera response varies by more than 5% across the centriod window, a flat fielding step must be added.

Conclusion

It should be noted that the algorithm in this report was written and presented in a manner which (I hope) makes it easy to understand. There are numerous modifications to the sequencing and method of computation which do not change the result but improve the efficiency of the algorithm significantly. The need for a more computationally efficient algorithm depends on the speed of the processor performing the computation. In order to track at the highest projected tracking rate of 1 frame/second with reasonable latency, this algorithm should be executed in less than 500 milliseconds.