Homework #4 Due 10/4

Read Book, Ch.8.

1. Return to the chart above, from HW#3, pictured above, and recall problem 4 of that homework:

“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].”

Referring to the chart below, note that the above exercise entailed identification of the fourth and sixth rows, based on information you collected in the lab 9/17.

This was just enough to solve for the four elements of the Jacobian, i.e. it is four equations for four unknowns. In an effort to use all of the collected data, fill in

all 24 places in this table, using the data you collected in the table at the top. Note that the ordering of the differences won’t actually matter to the eventual

estimates of elements of [J]; for the top left slot, for instance, it doesn’t matter whether you base Dq_pan on “q_4pan - q_1pan” or “ q_1pan - q_4pan ”. It is important to

preserve whichever order you choose in the remaining elements in that first row: Dq_tilt, Dx_c, Dy_c.

2 With the redundancy that this affords, we seek a least-squares best resolution of our four Jacobian elements. Thus we seek to minimize over all b :

J(b) = ½ r ^T [W] r = ½ S_i W_i r_i², i=1, 2, 3 … 12

where

r_2i-1 = Dx_cⁱ - [ b₁ Dqⁱ_pan+ b₂ Dqⁱ_tilt ]

r_2i = Dyⁱ_c - [ b₃ Dqⁱ_pan+ b₄ Dqⁱ_tilt ]

i = 1, 2, 3, 4, 5, 6 (i.e. each corresponding with a row of your table)

and where we have defined

J₁₁ = b₁ J₁₂ = b₂

J₂₁ = b₃ J₂₂ = b₄

Begin with an initial guess of b= [1 1 1 1]^T, and show that application of

Db = [A^TWA]^-1A^TWr(b_o)

, with W=I, results in the correct b with just one iteration. (Note that, this being a linear model, we would not ordinarily go to the trouble of applying Newton-Gauss, with its attendant “initial guess”.)

3 Please come to the lab at the usual times September 24 -- this time to acquire two images with the multiple spot at two different pan/tilt angles.  

Be sure the camera remains in a fixed location, not moving between the acquisitions of the two pictures.

Difference image two from image 1, then apply the mask of HW 3 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.

Write a simple program that will find most/all of the laser-spot centers in both images from this single differenced image.

Returning to your two initial images, replace the pixel center of each found spot in the respective images with a “zero”, and print out the

two consequent pictures.

Discuss in 100 words or so the way in which your program uses the conditioned/differenced image, as well as methods

you might apply to reduce the number of false negatives and/or false positives.