Exercises of PIV. incomplete draft, version 0.0. October 2009

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1 Exercises of PIV incomplete raft, version 0.0 October Images Images are signals efine in 2D or 3D omains. They can be vector value (e.g., color images), real (monocromatic images), complex or binary signals. 1. Consier a rectangular picture with imensions A B. Define two coorinate systems for this picture an obtain the geometric transformation between both coorinates systems. 2. Let I : D R be an image with omain D an coomain R. What can be sai aboutd an R if I is (a) iscrete image, (b) continuous image, (c) binary image, () monochromatic image, (e) color image, (f) vector value image. 3. Suppose you want to convert a finite image I with omain [0, A] [0, B] into an image with infinite omain R 2. Draw an efine three ways of oing this. 2 Image spaces Images live in image spaces. In many cases, images belong to vector spaces in which linear transformations are easily efine. 1. Let I, J be two real images with omain D R 2. Define the sum of the two images, I + J, an the multiplication of the image I by a scalar α R. 2. Show that the set of real images with omain D is a vector space. 3. Discuss if the set of integer images I : D Z is a vector space. 4. Consier the set of finite iscrete images of size M N. What is the imension of this space? Define one image basis for this set. (suggestion: assume M = 3, N = 2). 5. Let S be the set of iscrete real images with omain D. Prove that I J = I(m, n)j(m, n) (m,n) D verifies the properties of an inner prouct between the two images. (assume that the sum converges). 6. Define the norm of a iscrete image, inuce by the above inner prouct. 7. Let S be the space of real images with omain D = [0, A] [0, B]. If an image I S is rotate by an angle θ [ π, π[, oes the rotate image still belong to S? give an example. 1

2 3 Matrix notation Discrete images with finite omain can be represente by matrices an by vectors. This is very useful because many image processing operations can then be expresse using matrix calculus. 1. Consier a iscrete image I with omain D = {0, 1} {0, 1, 2, 3}, whose elements are efine by the table Represent this image as a 2 4 matrix an as a 8 1 column vector. 2. Consier a finite iscrete image represente by a matrix [ ] I = Convert this matrix into a column vector. Can this operation be performe by a left or right multiplication of I by an apropriate matrix A? 3. Suppose we want to a two finite iscrete images with the same omain D = {0,..., M 1} {0,..., N 1} how can this operation be efine if the images are represente by M N matrices. 4. Solve the same problem for the case of a multiplication of a matrix by a scalar. 4 Sampling an interpolation If we wish to store or manipulate an image in a computer we have to convert it into a iscrete image. On the contrary, if we want to isplay a iscrete image on an analogue TV or process it with supixel accuracy we have to convert it into a continuous image. The conversion between iscrete an continuous images is therefore neee. 1. Consier a continuous signal I : [0, A] R. Define the sampling operation with sampling interval T. 2. Convert the continuous signal I(x) = sin(πx), 0 x 5 into a iscrete signal, using a sampling interval (a) T = 1 (b) T = 0.5 (c) T = 0.25 In which cases o you think a perfect reconstruction is possible? why? 3. Convert the continuous signal I(x) = u(x 2), 0 x 5 (u is the unit step function) into a iscrete signal by sampling using a sampling interval (a) T = 1 (b) T = 0.5 (c) T = 0.25 In which cases a perfect reconstruction is possible? why? 4. Do you make any assumprions when you interpolate a signal? which assumptions i you make in the two previous problems? 2

3 5. Consier a iscrete signal of length 4 such that x(0) = 1, x(1) = 3, x(2) = 2, x(4) = 2. Define an interpolation algorithm to convert this iscrete signal into a continuous signal. Draw both signal. 6. Consier a continuous signal u(x) efine in the interval [0, N]. Define two vectors u,v by sampling the signal u(x) with rampling interval T = 1 an T = 1/2, respectively. (a) show that u = Hv where H is a (N + 1) (2N + 1) matrix. (b) is it possile to efine a matrix G such that v = Gu? 7. Given a finite iscrete signal represente by a vector u = [u 1... u N ] T efine two possible ways of interpolating the iscrete samples in orer to obtain a continuous signal u(x) such that u(i) = u i. Check if the interpolation algorithm preserves any class of signals such as constants, unit steps, polynomials, sinusois. 8. Check if the interpolation algorithm propose in the previous exercise can be written in the form u(x) = u i φ(x i) i where φ(x) is an interpolation function. 9. Given a iscrete image U = [u ij ], i {0,..., M}, j {0,..., N} efine two ways of interpolating the image in orer to obtain a continous image U(x, y), (x, y) [0, M] [0, N]. 10. Check if the interpolation algorithm propose in the previous exercise can be written in the form U(x, y) = U i,j Φ(x i, y j) i,j where Φ(x, y) is an interpolation function. 11. Suppose you wish to rotate a iscrete image. The rotation has amplitue θ an its center is the center of the image omain. Can you o this operation without interpolation? why? 12. Suppose you want to apply a geometric transform to an image (e.g., a rotation). Which of the following strategies are appropriate (a) transform each pixel of the input image an assign the intensity of the input to the output pixel. (b) for each pixel of the output image apply the inverse transform to fin the corresponing pixel in the input image. Then assign the intensity of the input to the output pixel 5 Geometric transformations 1. Write an expression for the rotation of a 2D point with amplitue θ an origin at O = (o x, o y ). Discuss the structure of the rotation matrix an the egrees of freeom. 2. Consier two geometric transformations in the plane: a rotation of amplitue θ followe by a translation t an vice-versa a translation t followe by a rotation θ. What is the relationship between (θ,t) an (θ,t ) if both transformations are ientical. 3. Consier the following transformations rigi boy: y = Rx + t R is [ a rotation ] matrix a b similarity: y = Bx + t B = b a affine: y = Ax + t A R 2 2 where x,y R 2. Do these transformations preserve parallel lines, angles an istances? what is the minimum number of points require to estimate the parameters of each transformation. 3

4 4. Let (x 1,y 1 ),..., (x K,y K ) be K pairs of points relate by an affine transform y= Ax+t. Derive an expression for A,t by minimizing the square error criterion E = K y i (Ax i + t) 2 i=1 Hint: use the property a 2 = a T a = tr{a T a} = tr{aa T } an see the erivative in appenix. 6 Filtering 1. Compare the convolution operations efine for iscrete an for continuous images. Fin the ifferences. 2. Ientify the ifference between the filter impulse response an the filter mask for the same filter. When are they equal? 3. Consier a 2D filter with impulse response h(x, y) = δ(x, y), where δ(x, y) = δ(x)δ(y) is the Dirac elta funcion. Show that, (a) δ(x, y) I(x, y) = I(x, y) (b) δ(x x 0, y 0 ) I(x, y) = I(x x 0, y y 0 ) 4. Repeat the previous exercise assuming that I is a iscrete image an h(m, n) = δ(m, n) is the iscrete impulse. Consier an input image I(m,n)=u(m) where u is the unit step function. Compute the output of the filter for the following masks (a) M = (origin at center of mask) (b) M = [ 1 0 1] T (origin at center of mask) (c) Sobel masks 5. Suppose the filter impulse response is a separable signal i.e., h(m, n) = h(m)h(n) an has length M N. Prove that the 2D convolution can be ecompose into M+N 1D convolutions. 7 Sinusois an Fourier transform 1. Consier a 2D sinusoi efine by φ(x, y) = cos(ω 1 x + ω 2 y), (a) characterize the level curves φ(x, y) = C, (b) what is the istance between level curves, (c) how is this istance relate to the vector of frequencies ω = [ω 1 ω 2 ] T. 2. Suppose you apply a complex exponential φ(x, y) = exp[j(ω 1 x+ω 2 y)] to the input of a linear filter with impulse response h(x, y). What is the output of the filter? 3. Repeat the previous problem assuming that the input an the filter impulse response are iscrete images. 4. Define frequency response of a 2D filter base on the results of previous exercises. 4

5 5. The Fourier transform of a iscrete image I is an image F{I} efine as follows F{I}(ω 1, ω 2 ) = (m,n) Z 2 I(m, n)e j(ω1m+ω2n) Prove that the Fourier transform of a iscrete image J(ω 1, ω 2 ) = F{I} is perioic in ω 1 an ω 2 with perio 2π, i.e., J(ω 1 + 2πp, ω 2 + 2πq) = J(ω 1, ω 2 ) 6. Compute the Fourier transform of the following signals (a) I(m, n) = δ(m, n) (b) I(m, n) = δ(m p, n q) (c) I(m, n) = (u(m + a) u(m a 1))δ(n) 7. The inverse Fourier transform of a spectrum J(ω 1, ω 2 ) is a iscrete image I(m, n) = F 1 {J(ω 1, ω 2 )} efine as follows I(m, n) = 1 (2π) 2 J(ω 1, ω 2 )e j(ω1m+ω2n) ω 1 ω 2 [ π,π] 2 8. Compute the inverse Fourier transform of the following signals (a) J(ω 1, ω 2 ) = δ(ω 1, ω 2 ) (b) J(ω 1, ω 2 ) = δ(ω 1 ω 01, ω 2 ω 02 ) 8 Image Reconstruction 1. (moving camera) Consier an ieal image I(x, y) an suppose this picture is capture by a moving camera. Assume that the camera takes t sec to acquire the image an uring this interval the camera moves in the x irection with velocity v x. Characterize the relationship between the observe image J(x, y) an the ieal image I(x, y). 2. Suppose an image I(m, n) was observe by a noisy sensor an the observe image is J(m, n) = I(m, n) + W(m, n) where W(m, n) is the noise introuce by the sensor. We wish to estimate the non-observe image I by minimizing a cost functional which measures the ifference between I(m, n) an J(m, n) E = (m,n) Z 2 (J(m, n) I(m, n)) 2 Minimize E with respect to I(m, n). Interpret the solution. 3. Consier the previous problem an assume that the cost function inclues a regularization (smoothness) term E = (J(m, n) I(m, n)) 2 +λ (I(m, n) I(m 1, n)) 2 +(I(m, n) I(m, n 1)) 2 (m,n) Z 2 (m,n) Z 2 Perform the minimization of E with respect to the unknown image I. Does the estimate of I(m, n) epen on its neighbors? 4. Suppose you want to solve an inpainting problem in which parts of the image are not observe (amage) an the others are observe without error. Coul you apply this technique to the inpainting problem? 5

6 5. If I, J are iscrete images with M lines an N columns, they can be represente by column vectors I, J. (a) Show that the cost function of the previous problem can be written as E = J I 2 + λ DI 2 where. is the Eucliean istance an D is an apropriate matrix. (b) Determine the stationary points of E by making its erivative equal to zero. (c) Can we guarantee that the secon erivative of E (Hessian matrix) is positive semiefinite an the stationary points are minima? 6. Repeat the previous problem for the case of a linear egraation moel (e.g., blurr) J = HI + W. Determine the cost function an minimize it. 9 Camera moel A camera projects 3D (worl) points into 2D (image) points. This is an important operation if we want to unerstan the content of the image an its relationship with the real worl. 1. Uner simplifying hypotheses the camera moel is given by the perspective projection x = f X Z y = f Y Z (1) where X, Y, Z are the coorinates a the original point in space an x, y the coorinates of the projecte point. (a) What are the hypothesis assume in this moel? (b) If a point moves in space with velocity v X (v Y = v Z = 0) what is the velocity of the point in the image plane? which points move fastest. (c) If a point moves along a straight line in space X = X 0 + αv what is the trajectory of the point in the image plane? 2. Suppose a point X lies on a plane α, parallel to the image plane. Derive a simplifie moel which projects points from the plane α into points on the image plane. Can this moel be use in practice? 3. Consier a more general moel for the camera λ x = P X P = K [R t] P R 3 4 (2) where X, x efine the original an projecte points in homogeneous coorinates, K is a upper triangular matrix of intrinsic parameters of the camera, R,t are extrinsic parameters of the camera (rotation matrix an translation vector) an λ is a scale factor. Uner which hypothesis is moel (1) equivalent to this one. 4. Knowing moel (2), etermine the coorinates of the optical center in terms of R,t. 5. Determine the optical center knowing the camera matrix P. 6. Consier a projective moel λ x = P X where P R 3 4. Express this moel using Cartesian coorinates. Suggestion: enote the lines of matrix P by π T i R Consier a set of parallel lines in 3D space X = X i +αv, i = 1,...,N. Project these lines on the camera plane an show that each projecte line has a vanishing point an all valishing points are equal. 6

7 8. Suppose you are watching a football match on a TV an suppose the images were obtaine with camera moelle by λ x = P X. Derive a 1-1 map between the position of the players in the fiel an in the image plane. 9. Supose you i not know the parameters of the camera in the previous problem an wante to estimate it from fiel observations. How many known points o you nee to measure (know) in the fiel? how woul you procee? 10 Optimization Many image processing problems can be formulate as optimization methos. This section focus a few usefuul techniques. 1. ( erivative with respect to a vector) Let f : R n R be a scalar function which maps a vector x into a scalar f(x). The erivative of f with respect to x (graient vector) is efine by f x = x 1. x n where x i enotes the i th component of x. Prove that (a) x at x = a, a,x R n (b) x xt Mx = (M + M T )x, x R n,m R n n 2. (erivative with respect to a matrix) Let f : R m n R be a function that maps a matrix X into a scalar f(x). The erivative of f with respect to matrix X is Show that f X = X X m1... (a) Xtr (AX) = AT (b) X tr ( AX T) = A (c) X tr (AXB) = AT B T () X tr ( AX T B ) = BA (e) X tr ( XX T) = 2X (f) X tr (AXBX) = AT X T B T + B T X T A T (g) X tr ( AXBX T) = A T XB T + BXA X 1m.. X mn 3. Consier a linear system of equations with more equations than unknowns: Ax = b where x R n,b R m an m > n. In most cases this problem is impossible since it is not possible to satisfy all the equations. However, the problem can be solve in an approximate way by minimizing the (square) norm of the error E = b Ax 2 Prove that the minimization of E is given by x = A # b where A # = A T (AA T ) 1 is the pseuo inverse of matrix A. 7

8 4. Suppose we wish to fin a vector in the null space of matrix A. This vector shoul be a solution of the equation Ax = 0. The trivial solution x = 0 is not interesting an shoul be eliminate. We will therefore impose an aitional constraint that x = 1. The problem can thus be formulate as follows minimize Ax 2 uner the constraint x 2 = 1 Prove that the optimal soution is the eigenvector associate to the smallest eigenvalue of matrix A T A. Hint: minimize the Lagrangean function L(x) = Ax 2 + λ( x 2 1). 8