Homework #3 Due 9/20

Read Book, pp 58–64 and Ch.7.

1. Please come to the lab at the usual times September 17 -- this time to acquire four images with the single spot at four different pan/tilt angles.  Be sure the camera remains in a fixed location, not moving between the acquisition of the four pictures.

Record the actual pan/tilt angles commanded in each of the four cases (note that these will not be in any recognizable units such as degrees,

and for this exercise there is no real need to know/convert); this you can get from Biao who will help you.  Then for homework use this information in order to create/test a Jacobian as per items 2-5 below.  There is no need for a particularly large separation in pan/tilt angles among the four cases -- anything on the order of ten degrees would be fine.  

2. Difference image two from image 1, then apply the mask indicated in order to condition this differenced image, replacing all pixel values,

except those in the rightmost, leftmost, uppermost, and lowermost 3 columns/rows with a new integer, based upon the mask formulation. Identify and enter into the table above the largest positive pixel in this result as the x_c y_c for Image 1. Identify the most negative as the x_c y_c for image two. Then repeat this for images 3 and 4, respectively..

3. Normalize the pixel values of the mask-conditioned image involving original images 1 and 2 by: (a) Dividing each pixel’s value by the highest magnitude of all conditioned values in the entire field and multiplying this quotient by 255, and (b) assigning the absolute value of the result to each pixel. Then print out a grayscale picture (similar to HW#1) of the result in the vicinity of the two laser spots.

4. Use rows 1 and 2 of your completed table above in order to find Dx_c Dy_c Dq_pan Dq_tilt

in the following approximation:

Dx_c = J₁₁Dq_pan+ J₁₂Dq_tilt

Dy_c = J₂₁Dq_pan+ J₂₂Dq_tilt

Similarly, use rows 1 and 3 to find other values of Dx_c Dy_c Dq_pan Dq_tilt

in

Dx_c = J₁₁Dq_pan + J₁₂Dq_tilt

Dy_c = J₂₁Dq_pan + J₂₂Dq_tilt

Use the four equations above to solve for approximate elements of the Jacobian of interest, [J].

5. Check out how good a job these J elements do at predicting, based upon the change from image 1 to image 4 of Dq_pan Dq_tilt, together with the row-1 (“starting”) x_c y_c , row-4’s x_c y_c . Note that in the presence of “feedback”, that is real-time sampling of the actual progress of the laser spot toward its user-selected destination, even very imperfect J elements can be used to produce fast convergence onto user-selected image junctures.