Homework 3 - SOLUTIONS

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1 EGGN 51 Computer Vson Sprng 13 Homeork 3 - SOLUTONS Due Monda, Februar 18, 13 b 8: pm Notes: Please emal me our solutons for these problems n order as a sngle Word or PDF document. f ou do a problem on paper b hand, please scan t n and paste t nto the document although ould prefer t tped! pts On the course ebste are to mages of a street ntersecton taken from a statonar camera, taken 5 seconds apart. Transform these mages to orthophotos ;.e., mages taken from a vepont drectl overhead, such that the scale s unform. Here are some control ponts that have been measured n the scene: mage, Actual, n feet 154, 389, 453, , -1 63, , -54.5, 5 5, , 464, , 436 8, , , Choose a scale of one pel =.5 feet n the output mage. Ho fast mles per hour s the person n the loer left alkng t s ok to measure the locaton of the person n the orthophotos b hand? Soluton: The to mages: The Matlab code: % HW3 p1 clear all close all 1

2 EGGN 51 Computer Vson Sprng 13 1 = mread'mage37.jpg'; msho1, []; % mage ponts, p1 = [ ; % Orgn ; % Stop 63 14; % Stop 5; % Lamp ; % Sdealk ; % Sdealk 7 31]; % Whte mark % Dra marks for =1:szep1,1 = p1,1; = p1,; rectangle'poston', [ ], 'EdgeColor', 'r'; end % Correspondng orld coordnates, n feet p = [ ; ; ; 5-56; 17.7; ; ]; % Scale so that one pel = S feet S =.5; p = p/s; T = cptformp1,p, 'projectve'; 1ortho = mtransform1, T, 'XData', [-1 7]/S, 'YData', [-7 3]/S; fgure, msho1ortho, []; mpelnfo % Poston of person's feet = 43, 37 measured b hand = mread'mage187.jpg'; fgure, msho, []; ortho = mtransform, T, 'XData', [-1 7]/S, 'YData', [-7 3]/S; fgure, mshoortho, []; mpelnfo % Poston of person's feet = 5, 3 measured b hand d = S*sqrt5-43^ ^; fprntf'person traveled %f feet n 5 second or %f mph\n',... d, d/5*.68; The orthophotos:

3 EGGN 51 Computer Vson Sprng 13 3 Person traveled feet n 5 second or mph. 15 pts The auto-correlaton score s AC E u u equaton 4.5 n the tetbook. Sho that ths equals u A u T, here matr A s as gven n equaton 4.8. Soluton: E AC u u

4 EGGN 51 Computer Vson Sprng pts Take the Matlab corner detector program developed n class and make the follong changes: a. nstead of usng a square regon of sze NN to sum the gradent product eght the value usng a Gaussan mask for,. b. nstead of usng the nterest pont measure deta/tracea, use the mnmum egenvalue of A 1. Ths s the Sh-Tomas approach. c. nstead of takng all nterest ponts above a mnmum threshold, take the 1 ponts th the hghest scores. Appl ths program to fnd the top 1 corner ponts n the mage test.jpg. Dra a rectangle around each of these ponts on the orgnal mage and label them. Soluton: The change for part a s straghtforard just use a Gaussan for. The sgma for the Gaussan s up to ou to pck. Our author states that sgma =. gves good results. For part b, e kno that the egenvalues of a smmetrc matr are a c a c b 4 v The smaller egenvalue ll be the one th the negatve sgn. You can compute the smallest egenvalue at each pont n the mage usng the Matlab command v = A11+A-sqrt A11-A.^ + 4*A1.^ /; For part c, the Matlab sort functon s hand. But ou not onl ant to return the sorted value ou ant to kno the ndces of the sorted ponts. You can do ths usng [val ndces] = sortval 'descend'; The Matlab code, and the resultng mage: clear all close all = doublemread'test.jpg'; % Appl Gaussan blur sd = 1.; = mflter, fspecal'gaussan', round6*sd, sd; msho, []; % Compute the gradent components G = mflter, [-1 1]; 1 Do not use for loops to go through the mage and call Matlab s eg functon at ever pont. nstead, use the equaton for the egenvalue that ou calculated n HW1, problem 3. You should be able to do ths thout an for loops. 4

5 EGGN 51 Computer Vson Sprng 13 G = mflter, [-1; 1]; % Compute the products of the gradents at each pel G = G.* G; G = G.* G; G = G.* G; % Sze of neghborhood over hch to compute corner features. N = 13; % = onesn; % The neghborhood s =.; % Sgma for ntegraton step = fspecal'gaussan', N, s; % The neghborhood % Sum the G's over the ndo sze. % Note: these convolutons can be epensve for large ndo szes. % f tme s crtcal, ou can alas do to 1D convolutons ro % frst, then column snce the mask s separable. For reall % large ndo do the convoluton n the Fourer doman. A11 = mflterg, ; A1 = mflterg, ; A = mflterg, ; % At each pel,, e have the matr % [A11, A1,; % A1, A,] % Of course, A1 = A1. % Fnd the egenvalues of A. These satsf the equaton A = v, here v % s an egenvalue and s the correspondng 1 egenvector. % We can solve b takng A - v =, and so e fnd v such that % deta - v =. % The to egenvalues actuall all e need s the v value % v1 = A11+A+sqrt A11-A.^ + 4*A1.^ /; v = A11+A-sqrt A11-A.^ + 4*A1.^ /; s = v; % Choose a suppresson radu for non-mama suppresson r = N; % Fnd local mama thn each neghborhood of radus r Lma = s==mdlate strel'dsk',*r; % Note - e don't ant to detect ponts too close to the border, so just % zero out everthng near the border. Lma1:N,: = false; Lma:,1:N = false; Lmaend-N:end,: = false; Lma:,end-N:end = false; % Get a lst of the ndces of all the potental nterest ponts [ros cols] = fndlma; 5

6 EGGN 51 Computer Vson Sprng 13 % Get the values of those nterest ponts vals = slma; % Sort n descendng order. "vals" are the sorted values; "ndces" are % the correspondng ndces of those values. [val ndces] = sortval 'descend'; % Dra a bo around the nterest pont of sze NN for =1:1 = colsndces; = rosndces; rectangle'poston', [-N/ -N/ N N],... 'EdgeColor', 'r',... 'Lnedth', 1.5; % default s.5 end tet+5,-5, sprntf'%d',,... 'Color', 'r',... % label th d number 'FontSze', 14; % default s pts Run the OpenCV verson of the Sh-Tomas corner detector, hch s mplemented n the functon goodfeaturestotrack. The use of the Sh-Tomas corner detector s llustrated n a tutoral on the ebste. a. Appl the program to the mage test.jpg and fnd the top 1 corners note - ou ma not get eactl the same corners as our Matlab program fnds n the prevous problem. Gve the code ou used and the resultng mage. b. Appl the program to the mage cube1.jpg. See f ou can fnd all the corners on the checkerboard pattern ou ll have change the program to ncrease the mamum alloable number corners to fnd. 6

7 EGGN 51 Computer Vson Sprng 13 Soluton: a just ran the tutoral from the ebste, ecept that used the mage test.jpg. Here s the result: And the code: /** goodfeaturestotrack_demo.cpp Demo code for detectng corners usng Sh Tomas method OpenCV team */ #nclude "opencv/hghgu/hghgu.hpp" #nclude "opencv/mgproc/mgproc.hpp" #nclude <ostream> #nclude <stdo.h> #nclude <stdlb.h> usng namespace cv; usng namespace std; /// Global varables Mat src, src_gra; nt macorners = 3; nt matrackbar = 1; RNG rng1345; const char* source_ndo = "mage"; /// Functon header vod goodfeaturestotrack_demo nt, vod* ; /** man */ nt man nt, char** argv { /// Load source mage and convert t to gra 7

8 EGGN 51 Computer Vson Sprng 13 src = mread "C:/Users/hoff/Documents/Teachng/Eggn51/h/h3/test.jpg", 1 ; cvtcolor src, src_gra, CV_BGRGRAY ; /// Create Wndo namedwndo source_ndo, CV_WNDOW_AUTOSZE ; /// Create Trackbar to set the number of corners createtrackbar "Ma corners:", source_ndo, &macorner matrackbar, goodfeaturestotrack_demo ; } msho source_ndo, src ; goodfeaturestotrack_demo, ; atke; return; /** goodfeaturestotrack_demo.cpp Appl Sh Tomas corner detector */ vod goodfeaturestotrack_demo nt, vod* { f macorners < 1 { macorners = 1; } /// Parameters for Sh Tomas algorthm vector<pontf> corners; double qualtlevel =.1; double mndstance = 1; nt blocksze = 3; bool useharrsdetector = false; double k =.4; /// Cop the source mage Mat cop; cop = src.clone; /// Appl corner detecton goodfeaturestotrack src_gra, corner macorner qualtlevel, mndstance, Mat, blocksze, useharrsdetector, k ; /// Dra corners detected cout<<"** Number of corners detected: "<<corners.sze<<endl; nt r = 4; for sze_t = ; < corners.sze; ++ { crcle cop, corners[], r, Scalarrng.unform,55, rng.unform,55, rng.unform,55, 1, 8, ; } 8

9 EGGN 51 Computer Vson Sprng 13 } /// Sho hat ou got namedwndo source_ndo, CV_WNDOW_AUTOSZE ; msho source_ndo, cop ; b For the cube1.jpg mage, changed the matrackbar parameter to allo up to corners. Here s the result: pts Consder the 33 template as shon

10 EGGN 51 Computer Vson Sprng 13 a. Compute b hand the normalzed cross correlaton score of template th mage f, at the center poston of f. You mght ant to check our anser usng Matlab s normcorr functon. f b. Gve a dfferent mage f such that the normalzed cross correlaton score of template th mage f, at the center poston of f, elds a score of -1. Assume that the mage s the tpe unsgned 8-bt nteger e, ts values le beteen and 55. Soluton: a The normalzed cross correlaton score of a template th mage f s c, t f t f t f t f t t t The mean of s zero. The sum of the squared values of s t t. We onl need to look at the center 33 porton of f. The mean of the 33 regon of f at the center locaton s.. Subtractng off the mean from that 33 regon of f results n The sum of the squared values t f 4 t f. The numerator s t f t f t. So c at the center pont s 18/sqrt4=.816. Ths matches hat normcorr produces. 1

11 EGGN 51 Computer Vson Sprng 13 f b An mage that s dentcal to the template, but has opposte sgns for each value, ll eld a cross correlaton score of -1.. Hoever, snce e are restrcted to the values that can be represented n unsgned 8-bt nteger e can t use negatve numbers. But the crosscorrelaton operator subtracts off the mean ana, so e can add a constant to the mage to make sure the values are non-negatve. So, f add 4 to the negatve of, get Dong a normalzed cross correlaton of th ths mage results n a score of -1. at the center. The other values of f don t matter snce the are outsde the boundares of the template. 6. pts Usng normalzed cross correlaton, match the top 1 ponts from the corner detector program of problem 3, from mage test.jpg to ther best matches n mage test1.jpg. You can use Matlab s normcorr functon. Mark the best matches n the second mage, and label them th ther dentfng nde from the frst mage.e., 1,, etc. Soluton: We append the follong code to the program from problem #3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % No match these ponts to another mage = doublemread'test1.jpg'; % Appl Gaussan blur = mflter, fspecal'gaussan', round6*sd, sd; fgure, msho, []; % For each corner pont found above, e ll etract a template submage of % sze NN centered on that pont, and tr to match t to the second mage. for =1:1 = colsndces; % Locaton of corner pont n mage 1 11

12 EGGN 51 Computer Vson Sprng 13 = rosndces; % Get the template from the frst mage, surroundng ths pont M = floorn/; T = -M:+M, -M:+M; C = normcorrt,; % Do normalzed cross correlaton % The scores mage C s bgger than, b M ros and M columns along % the sdes and the top and bottom. So hen e fnd the locaton of % the peak score, e should subtract M from the ndces. cma = mac:; [ ] = fndc==cma; = -M; = -M; fprntf'pont %d matches th score=%f\n',, cma; end rectangle'poston', [-N/ -N/ N N],... 'EdgeColor', 'r',... 'Lnedth', 1.5; % default s.5 tet,, sprntf'%d',,... 'Color', 'r',... % label th d number 'FontSze', 14; % default s 1 The output s: Pont 1 matches th score=.9988 Pont matches th score= Pont 3 matches th score= Pont 4 matches th score= Pont 5 matches th score= Pont 6 matches th score= Pont 7 matches th score= Pont 8 matches th score=.9689 Pont 9 matches th score=.9951 Pont 1 matches th score=

13 EGGN 51 Computer Vson Sprng 13 Lookng at the result these ponts matched correctl: 1,3,4,5,6,7,9,1. These ere rong:,8. t s nterestng to dspla the correspondng patches net to each other. n the fgure belo, the top ro are the nterest pont patches from mage one, and the bottom ro are the correspondng patches etracted from mage to. 13

14 EGGN 51 Computer Vson Sprng 13 %%%%%%%%%%%%%%%%%%% % Just out of curost, dspla the correspondng nterest pont patches. % Top ro ll be from mage one, bottom ro from mage to. fgure; for =1:1 = colsndces; % Locaton of corner pont n mage 1 = rosndces; % Get the template from the frst mage, surroundng ths pont M = floorn/; T = -M:+M, -M:+M; subplot,1,, mshot,[]; C = normcorrt,; % Do normalzed cross correlaton % The scores mage C s bgger than, b M ros and M columns along % the sdes and the top and bottom. So hen e fnd the locaton of % the peak score, e should subtract M from the ndces. cma = mac:; [ ] = fndc==cma; = -M; = -M; subplot,1,+1, msho-m:+m, -M:+M, []; end 14