Mid-term review ECE 161C Electrical and Computer Engineering University of California San Diego

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1 Mid-term review ECE 161C Electrical and Computer Engineering Univerity of California San Diego Nuno Vaconcelo Spring We have een in cla that one popular technique for edge detection i the Canny edge detector. It contain three parameter: a moothing parameter σ, and two threhold t 1 and t 2 that implement an hyterei mechanim which allow control over the likelihood of tarting v following a contour. Here we will jut worry with the threhold that i ued to tart a contour, the other one will remain contant. We denote the threhold that we allow to change by t. Figure 1 and 2 preent an image and three edge map obtained with different parameter etting. The combination of parameter that we ued to obtain the edge map are given by the following table. Eperiment t σ For each eperiment, indicate which i the reulting edge map. Provide an eplanation for your aignment, i.e. why did you aign edge map to eperiment y? (Note: the aignment themelve are only worth 10% of the grade in thi problem - make ure you provide a clear eplanation of your thought proce in determining them). 2. Conider the equence (n 1,n 2 ) whoe dicrete pace Fourier tranform i X(ω 1,ω 2 )=2+2co(w 1 ) 2co(w 2 ) a) Determine (n 1,n 2 ). b) I [n 1,n 2 ] a eparable equence? Suppoe that we can add or ubtract non-zero value to the (n 1,n 2 ) for which [n 1,n 2 ] = 0. What i the minimum number of entrie that need to be made nonzero in order to change the eparability of the equence (i.e. make it non-eparable if it i eparable or vice-vera)? Eplain why. c) Suppoe that we can make the non-zero value of [n 1,n 2 ] equal to zero. What i the minimum number of entrie that need to be made zero in order to change the eparability of the equence (i.e. make it non-eparable if it i eparable or vice-vera)?. Eplain why. 1

2 a) Image b) Edge map 1 Figure 1: 2

3 c) Edge map d) Edge map 3 Figure 2: 3

4 y y r P 1 r P 1 z z a) 2 + y 2 + z 2 = r 2 b) 2 + y 2 = r 2 y P 1 y z O 1 2 a b c 3 c) 2 + z 2 = y d) θ 1 =45,θ 2 =30,θ 3 =60 Figure 3: Lambertian urface with uniform albedo. 3. Figure 3, a) through d), how four Lambertian urface with uniform albedo. Each of thee urface i illuminated by a point ource at infinity (E = 1), the direction of which i indicated by the vector. The radiance reflected from the point P 0 on each urface i 10 unit. Determine the albedo of the urface and derive an epreion for the radiance reflected from a general point P 1 =(, y, z). Comment on the reulting illumination pattern for each cae. 4

5 4. Conider the problem of correcting a poorly hot photograph, uch a that hown in the left ide of figure 5. The problem i that the camera wa not properly oriented, and image feature do not appear in their natural orientation, e.g. the building do not appear to be vertical. We would like to correct the image, in order to obtain omething like the one hown on the right ide of the figure. We model the problem a hown below: the camera that took the picture wa ubject to a rotation (with repect to the ideally aligned camera ) of angle β around the optical ai. We let i, j, k be the coordinate ytem aociated with the ideal camera and i, j, k the one aociated with the real camera. Image plane j k i j (X,Y,Z) i k=k j 3D cene i Figure 4: Left: imaging et-up for problem 4. Right: the coordinate ytem of the real camera (i, j, k ), i a rotation of the coordinate ytem of the ideal camera (i, j, k). a) Show that the coordinate ytem are related by i = co(β) i in(β) j j = in(β) i +co(β) j k = k b) Conider a point of coordinate (X, Y, Z) in the old coordinate ytem. Uing a), write down the coordinate (X,Y,Z ) of the point in the new coordinate ytem a a function of (X, Y, Z). (Note: if you find yourelf inverting the ytem of equation in a) you are probably going in the wrong direction.) c) Let (, y) be the image coordinate of the perpective projection of the point (X, Y, Z) fortheold camera, and (,y ) the image coordinate of the perpective projection of (X,Y,Z ) under the new one. Derive the equation that determine (,y ) a a function of (, y). d) Uing the equation derived in c) it i poible to determine the impact of the tranformation on the ditance between any two image point. Conider two point a =( 1,y 1 )andb =( 2,y 2 ), and the point a and b into which they are mapped. What i the relationhip between a b 2 and a b 2? (Note: here we are talking about the Euclidean ditance, i.e. for a =(, y) a 2 = a T a = 2 + y 2.) 5

6 Figure 5: Left: poorly hot photograph. Right: deired image. 6