CS 231A Computer Vision Midterm

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1 CS 231A Computer Vson Mdterm Tuesday October 30, 2012 Set 1 Multple Choce (20 ponts) Each queston s worth 2 ponts. To dscourage random guessng, 1 pont wll be deducted for a wrong answer on multple choce questons! For answers wth multple answers, 2 ponts wll only be awarded f all correct choces are selected, otherwse, t s wrong and wll ncur a 1 pont penalty. Please draw a crcle around the opton(s) to ndcate your answer. No credt wll be awarded for unclear/ambguous answers. 1. (Crcle all that apply) If all of our data ponts are n R 2, whch of the followng clusterng algorthms can handle clusters of arbtrary shape? (a) k-means (b) k-means++ (c) EM wth a gaussan mxture model (d) mean-shft Only d 2. (Pck one) Suppose we are usng a Hough transform to do lne fttng, but we notce that our system s detectng two lnes where there s actually one n some example mage. Whch of the followng s most lkely to allevate ths problem? (a) Increase the sze of the bns n the Hough transform (b) Decrease the sze of the bns n the Hough transform (c) Sharpen the mage (d) Make the mage larger a 3. (Pck one) Whch of the followng processes would help avod alasng whle down samplng an mage? 1

2 (a) Image sharpenng (b) Image blurrng (c) Medan flterng where you replace every pxel by the medan of pxels n a wndow around the pxel (d) Hstogram equalzaton b 4. (Crcle all that apply) A Sobel flter can be wrtten as Whch of the followng statements are true = [ ] (1) (a) Separatng the flter, reduces the number of computatons by 30% (b) It s smlar to applyng a gaussan flter followed by a dervatve. (c) Separaton leads to spurous edge artfacts (d) Separaton approxmates the frst dervatve of gaussan. a and b 5. (Pck one) Whch of the followng are true for Egenfaces (PCA) (a) Can be used to effectvely detect deformable objects (b) Invarant to Affne Transforms (c) Can be used for lossy mage compresson (d) Is nvarant to shadows c 6. (Pck one) Downsamplng can lead to alasng because (a) Samplng leads to addtons of low frequency nose (b) Sampled hgh frequency components result n apparent low frequency components. (c) Samplng ncreases the frequency components n an mage. (d) Samplng leads to spurous hgh frequency nose b 7. (Pck one) If we replace one lens on a calbrated stereo rg wth a bgger one, what can we say about the essental matrx, E, and the fundamental matrx, F? (a) E can change due to the possble change n physcal length of the lens. F s unchanged. 2

3 (b) F can change due to the possble change n lens characterstcs. E s unchanged. (c) E can change due to the possble change n lens characterstcs. F s unchanged. (d) Both are unchanged. b 8. (crcle all that apply) Whch of the followng descrbes an affne camera but not a general perspectve camera? (a) Relatve szes of vsble features n a scene can be determned wthout pror knowledge (b) Can be used to determne the dstance from a object of a known heght (c) Approxmates the human vsual system (d) An nfntely long plane can be vewed as a lne from the rght angle. a, d The followng questons refer to ths pcture n Fgure 1 Fgure 1: Long hallway 9. (crcle all that apply) Assumng the hallway s very long and very straght, f an affne camera were drected down the hallway, whch condtons must be met to see anythng at the end of the hallway? (a) The camera s n the exact mddle of the hallway. (b) The camera s on the floor (c) The camera s mage plane must be very very large. (d) The vector representng the camera s vewng drecton s exactly parallel to the floor and the walls. D 10. (crcle all that apply) Whch of the followng characterstcs of the hallway shown can be determned usng an affne camera ponted straght down the hallway? 3

4 (a) The color/texture of the walls (b) The number of openngs n the hallway (c) If the hallway s gettng narrower (d) If the hallway s gettng wder C 4

5 2 True or False (20 ponts) True or false, brefly justfy your answer n one sentence or less. Correct answers are 2 ponts, -1 pont for each ncorrect answer. (a) (True/False) Fsherfaces works better at dscrmnaton than Egenfaces because Egenfaces assumes that the faces are algned. (False), both assume the faces are algned (b) (True/False) If you don t normalze your data to have zero mean, then the frst prncpal component found va PCA s lkely to be unnformatve. (True) (c) (True/False) Gven suffcently many weak classfers, boostng s guaranteed to get perfect accuracy on the tranng set no matter what the tranng data looks lke. (False), one could have two dataponts whch are dentcal except for ther label (d) (True/False) Boostng always makes your algorthm generalze better. (False), you can overft (e) (True/False) It s possble to blur an mage usng a lnear flter. (True) (f) (True/False) When extractng the egenvectors of the smlarty matrx for an mage to do clusterng, the frst egenvector to use should be the one correspondng to the second largest egenvalue, not the largest. (False), t s the largest one (g) (True/False) The Canny edge detector s a lnear flter. (False), It has non-lnear operatons, thresholdng, hysteress, non-maxmal supresson (h) (True/False) A zero skew ntrnsc matrx s not full rank because t has one less DOF. (True), It s stll full rank, even f t has one less DOF. () (True/False) A pcture of an outdoor feld wll never have a horzon wth an affne camera. (False), The feld s only of fnte length, and stll on earth, whch s approxmately sphercal, so t wll curve down relatve to the camera and occlude tself, whch yelds a horzon. 5

6 (j) (True/False) Any two cameras wth overlappng felds of vew can be used n a useful stereo rg when usng appearance-based features for correspondence. (False) If the cameras are facng each other, they can t see features on the surface, except on the edges whch could be hard to extract snce the backgrounds are dfferent from the two cameras, even f the object s the same. 6

7 3 Long Answer (40 ponts) 11. (10 ponts) Detectng Patterns wth Flters A Gabor flter s a lnear flter that s used n mage processng to detect patterns of varous orentatons and frequences. A Gabor flter s composed of a Gaussan kernel functon that has been modulated by a snusodal plane wave. The real value verson of the flter s shown below. g (x, y; λ, θ, ψ, σ, γ, ) = exp Where x = x cos (θ) + y sn (θ) y = x sn (θ) + y cos (θ) ( x 2 +γ 2 y 2 2σ 2 ) cos ( 2π x λ + ψ ) Fgure 2 shows an example of a 2D Gabor Flter Fgure 2: 2D Gabor Flter (a) (6 pont) What s the physcal meanng of each of the fve parameters of the Gabor flter, λ, θ, φ, σ, γ, and how do they affect the mpulse response? Hnt: The mpulse response of a gaussan flter s shown n Equaton 2, t s normally radally symmetrc, how would you make ths flter ellptcal? How would you make ths flter steerable? What does the 2D cosne modulaton do to ths flter? λ: θ: φ: σ: γ: gaussan (x, y) = 1 ( 2πσ 2 exp x2 + y 2 ) 2σ 2 (2) λ: represents the wavelength of the snusodal factor θ: represents the orentaton of the normal to the parallel strpes of a Gabor functon φ: phase offest σ: Sgma of the gaussan envelope γ: Spatal aspect rato, and specfes the ellptcty of the support of the Gabor functon 7

8 (b) (4 pont) Gven a Gabor flter that has been tuned to maxmally respond to the strped pattern n shown n Fgure 3, how would these parameters, λ 0, θ 0, φ 0, σ 0, γ 0, have to be modfed to recognze the followng varatons? Provde the values of the new parameters n terms of the orgnal values. Fgure 3: Reference Pattern. θ = θ 0 + π 4. ψ = ψ 0 + π 2 8

9 . θ = θ 0 + π 4, σ = 2σ 0, λ = 2λ 0 v. γ = 1 2 γ 0 9

10 12. (10 ponts) Stereo Reconstructon Fgure 4: Rectfed Stereo Rg x x (a) (4 pont) The fgure above shows a rectfed stereo rg wth camera centers O and O, focal length f and baselne B. x and x are the projected pont locatons on the vrtual mage planes by the pont P ; note that snce x s to the left of O, t s negatve. Gve an expresson for the depth of the pont P, shown n the dagram as Z. Also gve an expresson for the X coordnate of the pont P n world coordnates, assumng an orgn of O. You can assume that the two are pnhole cameras. Z = fb X = xz f (b) (6 pont) Now assume that the camera system can t perfectly capture the projected ponts locaton on the mage planes, so there s now some uncertanty about the pont s locaton snce a real dgtal camera s mage plane s dscretzed. Assume that the orgnal x and x postons now have an uncertanty of ±e, whch s related to dscretzaton of the mage plane. Gve an expresson of the X, Z locatons of the 4 ntersecton ponts resultng from the vrtual mage plane uncertanty. Gve an expresson for the maxmum uncertanty n the X and Z drectons of the pont P s locaton n world coordnates. All expressons should be n terms of mage coordnates only, you can assume that x s always postve and x s always negatve. Z mn = Z max = d = x x fb (x+e) (x e) fb (x e) (x +e) 10

11 Fgure 5: Rectfed Stereo Rg wth mage plane error 4e (d 2e)(d+2e) fb (x e) (x e) = fb x x f f f Z dff = Z max Z mn = fb Z md = X mn = (x e)z md X max = (x+e)z md X max X mn = Z md(2e) tem (? pont) Assumng the X coordnate of the pont P s fxed, gve an expresson of of the uncertanty of the reconstructon of Z wth n terms of the actual value of Z and the other parameters of the stereo rg. What s the depth uncertanty when Z s equal to zero? Fnd the depth when the uncertanty s at ts maxmum and gve a physcal nterpretaton and a drawng to explan. Z mn = fb (x+e) (x e) fb (x e) (x +e) Z max = d = x x Z dff = Z max Z mn = fb Z md = fb (x e) (x e) = fb x x X mn = (x e)z md f X max = (x+e)z md f X max X mn = Z md(2e) f 4x o (d 2e)(d+2e) 11

12 13. (13 ponts) AdaBoost algorthm (greedy tranng) Let f m (x) be the classfyng functon learnt after the m th teraton. f M (x) = M β m C m (x) (3) m=1 Where, C m (x) m {1,..., M} s a bunch of weak classfers learned n M teratons. C m : R { 1, 1}. We wll now look at a dervaton for the optmal β, C at the m th teraton gven β, C {1,..., m 1} ( N (a) (1 pont) (β m, C m ) = arg mn =1 L [y, f m 1 (x ) + βc(x )]), where L[y, g] = β,c exp( yg) s the loss functon. Show that (β m, C m ) can be wrtten n the form (β m, C m ) = arg mn β,c N =1 w (m 1) exp{ βy C(x )} (4) and defne w (m 1). Note that w (m 1) s the weght assocated wth the th data pont after m 1 teratons. β m, C m = arg mn = arg mn β,c β,c N =1 exp{ y f m 1 (x ) βy C(x )} N =1 w(m 1) exp{ βy C(x )} w (m 1) = exp{ y f m 1 (x )} (b) (3 pont) Express the optmal C m n the form argmn C Err(C), where Err(C) s an error functon and s ndependent of β. Err(C) should be defned n terms of the ndcator functon { I[C(x) y ] defned by 1, f C(x) y I[C(x) y ] = 0 f C(x) = y C m = arg mn e β y =C(x ) w(m 1) + e β y C(x ) w(m 1) C (e β e β ) N C m = arg mn C C m = arg mn C =1 w(m 1) N =1 w(m 1) I[y C(x )] I[y C(x )] + e β N =1 w(m 1) (c) (2 pont) Usng the C m from part (a), the optmal expresson for β m can be obtaned as 12

13 ( ) β m = 1 2 log 1 errm err m where err m = N =1 w(m) I[y C m(x )] N =1 w(m) Now, fnd the update equaton for w (m) From part(a), we have w (m) = w (m 1) e βmy C m(x ) Snce, y C m (x ) = 1 2I[y C m (x )] w (m+1) = w (m) e 2βmI[y C m(x )] e βm. and show that w (m) w (m 1) exp (2β m I[y C m (x )]) (d) (4 pont) We wll use ths algorthm to classfy some smple faces. The set of tranng mages s gven n Fg. 6. x s the face and y s the correspondng face label, {1, 2, 3, 4, 5}. x 1 x 2 x 3 y 1 = +1 y 2 = +1 y 3 = +1 x 4 x 5 y 4 = -1 y 5 = -1 Fgure 6: Tranng Set Faces We are also gven 3 classfer patches p 1, p 2, p 3 n Fg. 7. A patch detector I (+) (x, p j ) s defned follows: { I (+) 1, f mage x contans patch p j (x, p j ) = 1, otherwse I ( ) (x, p j ) = I (+) (x, p j ) p 1 p 2 p 3 Fgure 7: Classfer patches All classfers C m are restrcted to belong to one of the 6 patch detectors,.e. C m (x) {I (±) (x, p 1 ), I (±) (x, p 2 ), I (±) (x, p 3 )}. 13

14 If C 1 (x) = I (+) (x, p 2 ), w (0) = 1, {1, 2, 3, 4, 5} and β 1 = 1, what s the optmal C 2 (x)? What are the updated weghts w (1)? What s the fnal classfer f 2 (x) combnng C 1, C 2? Does I [f 2 (x) > 0] correctly classfy all tranng faces? C 2 (x) = I (+) (x, p 3 ) w (1) exp(2β 2 ) for = 1, 5 w (1) 1 for = 2, 3, 4 where, β 2 = 0.5log(1.5) f 2 (x) = C 1 (x) + 0.5log(1.5) C 2 (x) Yes, t correctly classfes all the tranng mages. 14. (10 ponts) Implct Shape Model (Applcaton of Generalzed Hough Transform for jont mage classfcaton and segmentaton) You are gven a dctonary D of 1000 patches extracted from an mage database. M(e, D) s a retreval operaton whch returns the closest dctonary patches from D for a query patch f. You are also provded tranng mages wth K descrptve mage patches f k extracted at dfferent locatons d k n each mage,.e you are gven {(f k, d k )} k=1,...,k. (a) (2 pont) Tranng: Let N L be the total number of locatons n an object measured from the object center. We wsh to produce a 1000 N L probablty matrx P, wth P(D, L j ) recordng the probablty of fndng a patch D D at a locaton L j wth respect to the object center x. All tranng mages are annotated wth the object center. Wrte smple pseudocode to fnd P(I, L j ) from the tranng mages n a manner smlar to Hough Transform.. Repeat followng steps for every patch (e, L) on a tranng mage, wth object center x Fnd the closest dsctonary patched usng G = M(e, D) Increment the bns n P correspondng to (G, L x) G G. Normalze P to obtan the probablty matrx (b) (3 pont) Testng (Classfcaton): Let {(e k, d k )} k=1,...,k be the set of patches extracted at dstnctve ponts from a Test mage at locatons d k. P (o, x e k, d k ) = P (o, x I, d k ).P (G e k ) (5) s the probablty of fndng the object o at center x gven the patch (e k, d k ). G {M(e k, D)} s the set of all matchng patches n D returned by M(e k, D). Smplfy P (o, x G, d k ) and show how the fnal terms are related to P computed n part (a). P (o, x G, d k ) = P (x o, G, d k )P (o G, d k ) (1 pont) P (o G, d k ) = P (o G ) d P(G, d) (1 pont) P (x o, G, d k ) P(G, d k x) (1 pont) 14

15 (c) (2 pont) Testng (Classfcaton): Gven P (o, x e k, d k ) for all patches n a test mage, provde an expresson for score(x, o) denotng the probablty of fndng the object centered at x. Assume P (e k, d k ) s a constant correspondng to a unform pror. score(x, o) k P (o, x e k, d k ) (d) (3 ponts) Now you are provded a segmentaton mask for objects durng tranng. Provde a method to obtan P ( pxel on object o, I, L), denotng the probablty of the pxel of a dctonary patch I located at locaton L (measured from the object center) beng on the object o. Mantan a segmentaton codebook for every patch I and locaton L. Each codebook entry corresponds to a pxel on the patch. Durng tranng for every patch I on the mage at a locaton L wth respect to center, fnd the pxels under the segmentaton mask and ncrement the correspondng pxel codebook entres. Fnally normalze all codebooks to obtan the probablty. 4 Short Answer (39 ponts) 15. (6 ponts) Parallel Lnes under Perspectve Transforms Fgure 8: Boxes rendered usng dfferent projectons 15

16 (a) (2 pont) The two boxes n Fgure 8 represent the same 3D shape rendered usng two projectve technques, explan ther dfferent appearance and the types of projectons used to map the objects to the mage plane. Fgure on the rght s an orthographc projecton, parallel lnes are parallel, fgure on the left s perspectve, parallel lnes at an angle to the camera plane have a vanshng pont (b) (2 pont) For each projecton, f the edges of the cubes were to be extended to nfnty, how many ntersecton ponts would there be? Left, none, rght, 3 vanshng ponts. (c) (1 pont) What s the maxmum number of vanshng ponts that are possble for an arbtrary mage? There s no lmt, any set of parallel lnes at an angle to the camera wll converge at a vanshng pont (d) (1 pont) How would you arrange parallel lnes so that they do not appear to have a vanshng pont? Place lnes that are parallel to the camera plane, they wll converge at the pont at nfnty 16. (6 ponts) Cascaded Hough transform for detectng vanshng ponts Y AXIS X AXIS Fgure 9: Hough Transform For ths problem we are gong to use the slope m ntercept c representaton of lne: y = mx + c. The attached Fgure 9 shows the vertces of a rectangular patch under a perspectve transformaton. We wsh to fnd the vanshng pont n the mage usng a Hough Transform. 16

17 (a) (2 pont) Plot the Hough Transform (for lne fttng) representaton of the mage. Assume no bnnng and make plots n a contnuous (m, c) space. Just show the ponts wth two or more votes. See fgure Intercept (c) slope (m) Fgure 10: Make your plot here (b) (2 pont) Now usng y = mx + c representaton, run Hough transform on the results from part (a) (after usng a threshold of 2 votes) to get a representaton n the (x, y) space agan. The same plot as the plot n the problem statement, wth an ntersecton at (2.5, 1.75) (c) (2 pont) Explan how to extract the vanshng pont from the representaton n part (b). The vanshng pont s (2.5, 1.75) 17. (5 ponts) Usng RANSAC to fnd crcles Suppose we would lke to use RANSAC to fnd crcles n R 2. Let D = {(x, y )} n =1 be our data, and let I be the random seed group of ponts used n RANSAC. (a) (2 ponts) The next step of RANSAC s to ft a crcle to the ponts n I. Formulate ths as an optmzaton problem. That s, represent fttng a crcle to the ponts as a problem of the form mnmze L(x, y, c x, c y, r) I where L s a functon for you to determne whch gves the dstance from (x, y ) to the crcle wth center (c x, c y ) and radus r. 17

18 L(x, y, c x, c y, r) = sqrt((x c x ) 2 + (y c y ) 2 ) r (b) (2 ponts) What mght go wrong n solvng the problem you came up wth n (1) when I s too small? The problem s underdetermned. Wth e.g. 2 ponts there are nfntely many crcles one can ft perfectly. (c) (1 pont) The next step n our RANSAC procedure s to determne what the nlers are, gven the crcle (c x, c y, r). Usng these nlers we reft the crcle and determne new nlers n an teratve fashon. Defne mathematcally what an nler s for ths problem. Menton any free varables. An nler s a pont (x, y) such that sqrt((x c x ) 2 + (y c y ) 2 ) r T for some threshold T. 18. (6 ponts) Fast forward camera Fgure 11: Camara movement Suppose you capture two mages P and P n pure translaton n the Z drecton shown n Fgure 11. Image planes are parallel to the XY plane. (a) (3 ponts) Suppose the center of an mage s (0,0). For a pont (a, b) on mage P, what s the correspondng eppolar lne on mage P? the lne goes through (0, 0) and (a, b) on mage P. (b) (3 ponts) What s the fundamental matrx n ths case? (6 ponts) K-Means 18

19 Fgure 12: A wld jackalope (a) (3 pont) What s lkely to happen f we run k-means to cluster pxels when we only represent pxels by ther locaton? Wth k=4, draw the boundary around each cluster and mark each cluster center wth a pont for some clusters that mght result when runnng k-means to convergence. Draw on Fgure 12, and set the ntal cluster centers to be the four corners of the mage. Just a bunch of (roughly) equal-szed clusters tlng the plane, dependng on the startng locatons. For the mage, we expect each quadrant to be ts own cluster. (b) (1 pont) What does ths tell us about usng pxel locatons as features? It s not suffcent, we need rcher features. (c) (2 pont) We replace the sum of squared dstances of all ponts to the nearest cluster center crteron n k-means wth sum of absolute dstances of all ponts to the nearest cluster center,.e. our dstance s now gven by d(x 1, x 2 ) = x 1 x 2 1. How would the update step change for fndng the cluster center? At every teraton 1. Assgn ponts to the closest cluster center 2. Update the cluster centers wth the medan of ponts belongng to a cluster 20. (6 ponts) Canny Edge Detector (a) (3 ponts) There s an edge detected usng the Canny method. Ths detected edge s then rotated by θ as shown n Fgure 13, where the relaton between a pont on the orgnal edge (x, y) and a pont on the rotated edge (x, y ) s gven by x = x cos θ (6) y = x sn θ (7) 19

20 Wll the rotated edge be detected usng the Canny method? Provde ether a mathematcal proof or a counter example. Our rotaton s gven by Fgure 13: Edge Rotated by θ x = x cos θ y = x sn θ Our canny edge depends on the magntude of the dervatve whch s the only part of the algorthm whch could have really changed. Ths s gven by Dx 2 x + D 2 y y = cos 2 θd 2 xx + sn 2 θdyy 2 = D 2 xx whch s the same rule for the orgnal edge thus we have shown that the Canny method s rotatonally nvarant. (b) (3 ponts) After runnng the Canny edge detector on an mage, you notce that long edges are broken nto short segments separated by gaps. In addton, some spurous edges appear. For each of the two thresholds (low and hgh) used n hysteress thresholdng, state how you would adjust the threshold (up or down) to address both problems. Assume that a settng exsts for the two thresholds that produces the desred result. Explan your answer very brefly. The gaps n the long edges requre a lower low threshold: parts of the long edge are detected, so the hgh threshold s low enough for these edges, but the edges are dsconnected because the low threshold s too hgh. Lowerng the low threshold wll nclude more pxels of the long edges. Elmnatng the spurous edges requres a hgher hgh threshold. The hgh threshold should be ncreased only slghtly, so as not to make the long edges dsappear. The assumpton n the problem statement ensures that ths s possble. 20

21 21. (4 ponts) Image Segmentaton Both the mnmum cut and the mnmum normalzed cut algorthms for mage segmentaton have problems relatng to the sze of the parttons that they produce. (a) (2 pont) What s the problem wth the mnmum cut formulaton, and how s ths dealt wth by the mnmum normalzed cut formulaton? Mn cut prefers very small and very large parttons (b) (2 pont) Although usng normalzed cuts s much better than usng mnmum cuts, t stll has a problem concernng the sze of the parttons t produces. In what way s t based? Prefers parttons of roughly equal sze 21

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