Camera system: pinhole model, calibration and reconstruction

Transcription

1 Camera system: pinhole model, calibration and reconstruction Francesco Castaldo, Francesco A.N. Palmieri December 22, 203 F. Castaldo and F. A. N. Palmieri are with the Dipartimento di Ingegneria Industriale e dell Informazione, Seconda Università degli Studi di Napoli, Aversa (CE), Italy

2 Contents Introduction 2 2 Pinhole camera model D-3D geometry D-2D geometry Camera calibration 9 3. Projection matrix Sensitivity of projection matrix Homography Sensitivity of homography World point reconstruction D world point reconstruction Intersection Triangulation D world point recostruction Intersection Triangulation

3 Chapter Introduction This report investigates the functioning of a camera, analyzing its matematical model, the calibration stage and the world point reconstruction [3] [7] [4]. A generic camera system allows to reconstruct the position of a point in a three-dimensional (3D) space (the so-called world point), given the bi-dimensional (2D) point s projections on the cameras image planes (also called pixel projections) [3]. However, very often we deal with objects that are constrained to move in 2D spaces, e.g. ships moving on the sea plane for target tracking [5] (Figure. left). Therefore, we could simplify the camera model by assuming that the world points always lie on a plane. This situation defines an homography [3], that is a mapping between spaces with the same number of coordinates (2D world points and 2D pixels). In the following for each component of the camera we will address both the 2D-3D (2D pixels - 3D world points) and 2D-2D (2D pixels - 2D world points) scenarios. The 2D-D case is not very common and it will be neglected in this report. Figure.: On the left there is a scenario in which the 2D-2D hypothesis holds (in target tracking for harbour scenarios the ships are bounded to move onto the sea plane). On the right a scenario in which we have to use the 2D-3D model (airplanes fly in a 3D world space). 2

4 Chapter 2 Pinhole camera model In this chapter we present the camera model for a generic set up (3D-2D), and then deal with the 2D-2D case. 2. 2D-3D geometry A camera is a mapping between 3D world objects and a 2D image. We consider the simplest camera model, the so-called pinhole model [3]. From Figure 2. point C, to which all viewing rays from 3D space points converge, is the origin of the Euclidean coordinate system and it is called the center of projection. The center of projection is also called the camera center or the optical center. This model emerges as the equivalent of a classical camera setup in which the camera plane (also called image plane) is behind the camera center C. The image that is formed upside down on the camera plane is transposed in front of the camera center where the image appears in its natural orientation and with the same dimensions. Such a plane Z c = f is the new image plane, as shown in Figure 2.. The line that heads perpendicularly from the camera center to the image plane is called the principal axis for the camera. The intersection p of the principal axis and the image plane is called principal point. A 3D point X c = (X c, Y c, Z c ) T in the Euclidean space (X c, Y c, Z c ) centered in C (called the camera standard reference frame) is mapped to a point x = (x, y) T on the image plane where the viewing line of X c meets the image plane. The image reference frame is centered in p with x and y axes parallel to X c and Y c. Considering the plane (Y c, Z c ) as shown in Figure 2.2, by similarity of (ĈZ c Y c ) and (Ĉzy), we can write y = fy C Z C. By the same reasoning, in the plane (X c, Z c ) we have x = fx C Z C. Therefore the point (X C, Y C, Z C ) T is mapped to the point (fx C /Z C, fy C /Z C, f) T, that lies on the image plane. Ignoring the obvious third coordinate, we can write ( X c = (X c, Y c, Z c ) T x = (x, y) T fxc =, fy ) T c (2.) Z c Z c 3

5 viewing line camera centre principal axis principal point image plane Figure 2.: Pinhole camera model with image plane parallel to (X c, Y c ) Note that we cannot write (2.) in terms of matrix multiplications, due to the non-linearity of the mapping between the coordinates of X c and x. It is convenient (as it has become standard in the literature) to define world and image points by the augmented vectors X c = ( Xc ), x = ( x called homogeneous coordinates [3]. Observing that fx c f fy c = 0 f 0 0 Z c ), X c Y c Z c, (2.2) multiplying and dividing the first member of (2.2) by Z c we can write Z c fx c Z c fy c Z c = Z c x, and then write X c Z c x = fx c fy c Z c = f f X c Y c Z c. 4

6 Figure 2.2: Relation of coordinates system to focal length More compactly, we have λx = P X c, (2.3) where λ is a generic scale factor that depends on the specific point X c. Another way to write this expression is x P X c, where denotes equalities up to a scale factor. Matrix f P = 0 f 0 0, is called the camera projection matrix. The camera model just presented above is the simplest one and it is idealized because we have assumed that the origin of coordinates in the image plane is at the principal point and that the image plane is parallel to the (X c, Y c ) plane. However, cameras deployed in arbitrary position and orientation need to be described in a more general reference system. A first generalization assumes that the origin of image coordinates is not in in the principal point p. Denoting p = (p x, p y ) T, we can write X c x = (fx c /Z c + p x, fy c /Z c + p y ) T, that in homogeneous coordinates is X c λx = fx c + Z c p x fy c + Z c p y Z c = P X c Y c Z c, (2.4) where P = f 0 p x 0 0 f p y = K[I 3 0], 5

7 Figure 2.3: World to camera coordinate frames with K = f 0 p x 0 f p y 0 0 and I 3 the 3 3 identity matrix. K is called the camera calibration matrix. Generalizing the standard camera coordinate frame (X c, Y c, Z c ) to generic coordinates, known as the world coordinate frame (X, Y, Z), as shown in Figure 2.3, we can relate the two coordinate frames through a rotation matrix R and a translation vector t. Therefore X c = R X Y Z C x C y C z, = R( X C), where X = (X, Y, Z) T and C = (C x, C y, C z ) T represent respectively the generic 3D point and the camera center in the world reference frame. In homogeneous coordinates we can write X c = [ R R C 0 ] X Y Z [ R = R C 0 ] X, (2.5) ( ) x where x = is the generic point in homogeneous image coordinates. Therefore, putting (2.4) and (2.5) together, we have λx = P X, (2.6) with P = KR[I 3 C]. We can decide not to show explicitly the camera center C and write the transformation as X c = R X + t, with t = R C. Therefore 6

8 the 3 4 camera projection matrix is P = K[R t] Another element that we should take into account is the possibility of different scaling along both image coordinates. In the model derived above we assumed that image coordinates are Euclidean with equal scales in both axes directions. In transferring the coordinates to pixels, we will need to multiply the original coordinates by scale factors m x and m y as pixels may have different equivalent dimensions in x and y (the number of pixel per unit distance in x and y direction may be different). Matrix K becomes K = α x 0 x 0 0 α y y with α x = fm x and α y = fm y and x 0 = m x p x and y 0 = m y p y the coordinates of the principal points in terms of pixels in x and y direction. In the form derived above the parameters contained in K {f, α x, α y, x 0, y 0 } are called internal or intrinsic parameters, and the parameters in R and t are called external or extrinsic parameters. This camera model has 0 degrees of freedom ({α x, α y, x 0, y 0, f} form a set of four independent parameters plus we have three for R and three for t). It is useful to calculate the camera center C = (C x, C y, C z, ) T, that is the -dimentional right null-space of P, i.e. P C = 0. Algebraically, the center may be obtained with the following equations Cˆ x = det([p 2, p 3, p 4 ]), Ĉ y = det([p, p 3, p 4 ]), Ĉ z = det([p, p 2, p 4 ]), Ĉ t = det([p, p 2, p 3 ]), (2.7) where det( ) denotes the determinant of the matrix in the brackets, p i, i =, 2, 3, 4 denotes the columns of P and with C ˆ x C x /Ĉt C = C y C z = Ĉ y /Ĉt Ĉ z. (2.8) /Ĉt Ĉ t /Ĉt It is important in many application to calculate, given a camera C i and a pixel x on the camera image plane, the ray on which lie the (infinite) 3D points that can generate that projection x. This 3D rect can be written using two known points on the ray. The first is the camera center C, the second is the point X +, calculated as λ X + = P + x (2.9) where P + denotes the Moore-Penrose pseudo-inverse of P. Therefore the 3D rect can be written as X(β) = X + + βc (2.0) where β can vary arbitrary and determines the 3D points on the ray. 7

9 2.2 2D-2D geometry If world points are bounded to move on a plane, equation (2.6) becomes where x = (x, y, ) T, X = (X, Y, ) T, and h h 2 h 3 H = h 2 h 22 h 23, h 3 h 32 h 33 λx = HX, (2.) is a 3 3 homography matrix. The relationship between camera projection matrix P and homography H is the following. If P = [p, p 2, p 3, p 4 ], where p i, i =, 2, 3, 4 denotes the column of P, then H = [p, p 2, p 4 ]. 8

10 Chapter 3 Camera calibration In this section we address the camera calibration for both projection matrices and homographies. 3. Projection matrix The goal of camera calibration is to calculate the camera projection matrix P defined above when the sensor is deployed in arbitrary locations. We assume to know a certain number of point correspondences {x i, X i }, i =,.., N between 3D points X i and 2D image points x i (projections on the image plane of X i ), and we are required to find the matrix P such that λx i = P X i, as shown in (2.6), for all i. World points may be extracted from known landmarks or objects. There are several ways to accomplish calibration. One possible approach [7] is to rewrite equation (2.6) as where and λx i λy i λ P = = p p 2 p 3 p 4 p 2 p 22 p 23 p 24 p 3 p 32 p 33 p 34 λx i λy i λ = λx i = λ p p 2 p 3 p 4 p 2 p 22 p 23 p 24 p 3 p 32 p 33 p 34 We can re-write (3.) as { xi = λxi λ = pxi+p2yi+p3zi+p4 p 3X i+p 32Y i+p 33Z i+p 34 y i = λyi λ x i y i X Y Z, (3.) (3.2) = [p ij ]. (3.3) = p2xi+p22yi+p23zi+p24 p 3X i+p 32Y i+p 33Z i+p 34 i =,..., N. 9 (3.4)

11 We can write a homogeneous linear system using (3.4) for each known correspondence. In our model P has 0 degrees of freedom, therefore we need at least 5 world-images point matches, but in general, using a calibration pattern, we can obtain many more correspondences and consequently calculate a more accurate solution through a least-squares method. In fact, given N correspondences, we can write with and A = Ap=0, X Y Z x X x Y x Z x X Y Z y X y Y y Z y X 2 Y 2 Z x 2 X 2 x 2 Y 2 x 2 Z 2 x X 2 Y 2 Z 2 y 2 X 2 y 2 Y 2 y 2 Z 2 y X N Y N Z N x N X N x N Y N x N Z N x N X N Y N Z N y N X N y N Y N y N Z N y N p = [p, p 2,..., p 33, p 34 ] T. We have to solve a set of homogeneous equations in which there are more equations than unknowns. The obvious solution p = 0 is not of interest, therefore we have to pick a constraint in order to impose the minimization problem. A reasonable constraint would be p = ; we also observe that if p is a solution, then so is kp for any scalar k. Therefore the least-squares problem can be stated as finding p that minimizes Ap subject to p =. As showed in [3], using the singular values decomposition (SV D) [2] on A, we get A = UΣV T, then the problem requires to minimize UΣV T p. However, UΣV T p = ΣV T p and p = V T p (using the properties of orthonormal matrices U and V ). Therefore, we have to minimize ΣV T p subject to V T p =. If we put y = V T p, the problem becomes to minimize Σy subject to y =. Since Σ is a diagonal matrix with its elements in descending order, the minimum solution is simply y = (0, 0,..0, ) T, and since p = V y is simply the last column of V. Therefore, if A = [a ij ] is a 2N 2 matrix, with N > 5, then A = UDV T (3.5) is the economy size SV D of A, with U = [u ij ] and V = [v ij ] two orthogonal matrices 2N 2 and 2 9 respectively, and D = [d ij ] a 2 2 matrix containing the singular values of A. Using the procedure detailed above, p can be calculated as p = [v,2, v 2,2, v 3,2, v 4,2, v 5,2, v 6,2, v 7,2, v 8,2, v 9,2, v 0,2, v,2, v 2,2 ] T, (3.6) where v i,2, i =,.., 2 denotes the elements of the last column of V. Rearranging p we get P. Improvements of the resulting P can be achieved implementing data normalization and iterative procedures to reduce the error [3]. 0

12 3.. Sensitivity of projection matrix In this section we assess the impact of a noisy calibration stage to reconstruction quality. We assume that world points for the calibration have been perfectly estimated, while pixel correspondences are burdened with a variable quantity of noise. We refer to the procedure detailed in [6] [] for projection matrices. We calculate the partial derivatives of the elements of P = [p kl ], k =, 2, 3, l =, 2, 3, 4, respect to the calibration world points and pixels, defined as (X s, Y s, Z s ) and (x s, y s ), s =,.., N respectively. From (3.22) and (3.6) the following relations [] are obtained p kl x s = i,j p kl x s = X s v 3(k )+l,2 a 2s,9 v 3(k )+l,2 Y s a 2s,0 Z s v 3(k )+l,2 a 2s, v 3(k )+l,2 a 2s,2, (3.7) and p kl y s = i,j p kl y s = X s v 3(k )+l,2 a 2s,9 v 3(k )+l,2 Y s a 2s,0 Z s v 3(k )+l,2 a 2s, v 3(k )+l,2 a 2s,2, (3.8) To compute the partial derivatives (3.7) and (3.8) we need the partial derivatives of V respect to the entries of A. As showed in [6], by taking the derivative of (3.5) respect to a ij, we obtain A = U DV T + U D V T + UD V T (3.9) Since V is an orthogonal matrix, we have that where V T V = I = V T V + V T V = Ω ij V + ΩijT V (3.0) Ω ij V = V T V = [Ω ij V k,l ] (3.) is by definition a 2 2 skew-symmetric matrix. The same can be done for U, with Ω ij U = U T U = [Ω ij U k,l ] again a 2 2 skew-symmetric matrix. From (3.9), multiplying by U T and V from the left and right respectively, we get U T A V = Ω ij U D + D + DΩ ij V (3.2)

13 In order to compute the derivatives of v ij, we need to know Ω ij V, the elements of which can be calculated by solving a set of 2 2 linear systems in the form d l Ω ij U k,l + d k Ω ij V k,l = u ik v jl, d k Ω ij U k,l + d l Ω ij V k,l = u il v jk, (3.3) deduced by the antisymmetric properties of Ω ij U and Ωij V, with k =,.., N, l = i +,.., N. Once Ω ij has been computed, the sought derivatives of V are simply V V = V Ω ij V. (3.4) We use (3.4) to finally compute (3.7) and (3.8) for all the calibration pixels. These derivatives can be combined into the total differential dp kl = p kl x dx + p kl y dy p kl x N dx + p kl y N dy, (3.5) that sums the contributions of all the pixel errors for a single entries of P. The dx, dy,.., dx N, dy N values can vary accordingly to the estimated quality of the calibration process. In this way we can measure the uncertainty of the P matrix, defined in matrix form as P = 3.2 Homography dp dp 2 dp 3 dp 4 dp 2 dp 22 dp 23 dp 24 dp 3 dp 32 dp 33 dp 34 = [ p ij ]. (3.6) We detail the procedure for finding H. Given the correspondences {x i, X i }, i =,.., N between 2D world points X i and 2D image points x i we can re-write (2.) as λx i h h 2 h 3 X λy i = h 2 h 22 h 23 Y, (3.7) λ h 3 h 32 h 33 where and λx i λy i λ H = = λx i = λ h h 2 h 3 h 2 h 22 h 23 h 3 h 32 h 33 Equation (3.7) can be re-written as x i = λxi λ = hxi+h2yi+h3 h 3X i+h 32Y i+h 33, y i = λyi λ x i y i = h2xi+h22yi+h23 h 3X i+h 32Y i+h 33, i =,..., N. 2 (3.8) = [h ij ]. (3.9) (3.20)

14 Finding H can be reduced to the solution of the homogeneous linear system with and A = Ah = 0, (3.2) X Y x X x Y x X Y y X y Y y X 2 Y x 2 X 2 x 2 Y 2 x X 2 Y 2 y 2 X 2 y 2 Y 2 y X N Y N x N X N x N Y N x N X N Y N y N X N y N Y N y N h = [h, h 2, h 3, h 2, h 22, h 23, h 3, h 32, h 33 ] T. (3.22) We need at least 4 world-pixel matches, because each correspondence provides two independent equations and H has rank 8. With more than 4 correspondences, the set of equations is over-determined and we need an approximate solution, namely a vector h that minimizes a cost function. We are not interested in the obvious solution h = 0, therefore we need an additional constraint, that can be h = (H is defined up to a scale factor, therefore the value of the norm is unimportant). In this way, the solution h is obtained by minimizing the norm Ah, subject to h =. As showed in [3], we can recover the least-squares solution by applying singular values decomposition (SV D) [2] on A, with h that becomes the unit singular vector corresponding to the smallest singular value of A. If A = [a ij ] is a 2N 9 matrix, with N > 4, then, as before, A = UDV T is the economy size SV D of A, with U = [u ij ] and V = [v ij ] two orthogonal matrices 2N 9 and 9 9 respectively, and D = [d ij ] a 9 9 matrix containing the singular values of A. Using the procedure detailed above, h can be calculated as h = [v,9, v 2,9, v 3,9, v 4,9, v 5,9, v 6,9, v 7,9, v 8,9, v 9,9 ] T, (3.23) where v i,9, i =,.., 9 denotes the elements of the last column of V. Rearranging h we get H Sensitivity of homography We apply the same procedure detailed in [6] [] to assess the sensitivity for the homographies. We calculate the partial derivatives of the elements of H = [h kl ], k =, 2, 3, l =, 2, 3, respect to the calibration world points and pixels, defined as (X s, Y s ) and (x s, y s ), s =,.., N respectively. From (3.22) and (3.23) the following relations [] are obtained h kl x s = i,j h kl x s = X s v 3(k )+l,9 a 2s,7 Y s v 3(k )+l,9 a 2s,8 v 3(k )+l,9 a 2s,9, (3.24) 3

15 and h kl y s = i,j h kl y s = X s v 3(k )+l,9 a 2s,7 Y s v 3(k )+l,9 a 2s,8 v 3(k )+l,9 a 2s,9. (3.25) To compute the partial derivatives (3.24) and (3.25) we need the partial derivatives of V respect to the entries of A. We use the same procedure detailed for the 3D-2D case. We calculate (3.4) and compute (3.24) and (3.25) for all the calibration pixels. These derivatives can be combined into the total differential dh kl = h kl x dx + h kl y dy h kl x N dx + h kl y N dy, (3.26) that sums the contributions of all the pixel errors for a single entries of H. The dx, dy,.., dx N, dy N values can vary accordingly to the estimated quality of the calibration process. In this way we can measure the uncertainty of the H matrix, defined in matrix form as H = dh dh 2 dh 3 dh 2 dh 22 dh 23 dh 3 dh 32 dh 33 = [ h ij ]. (3.27) 4

16 Chapter 4 World point reconstruction In this section we address the problem of world point localization, given the projections of the point into the cameras image planes. As usual, we detail the procedure for both 2D-3D and 2D-2D cases. 4. 3D world point reconstruction Assuming that projection matrices P i are known for each camera i, we can localize a generic 3D point X = (X, Y, Z, ) T using camera matrices and corresponding 2D points x i = (x i, y i, ) T on each camera, that are the projection of the same X onto the two image planes. A separate problem is the identification of corresponding points in different images (i.e. given one pixel in the first image find automatically its homologous in the second image) [7]. In this paper we do not address such a problem, and in the experiments that will follow we have manually localized homologous points in pairs of images. Object identification is a important larger problem that we will address elsewhere. 4.. Intersection If we have available only two cameras, the simplest resolution of the problem, also known as camera intersection [4], can be stated as follows. From (2.6) applied to both cameras, we have λ x = P X λ 2 x 2 = P 2 X. This pair of equations can be rearranged in matrix form as AX λ = 0, (4.) 5

17 with and A = [ P x 0 P 2 0 x 2 X λ = X Y Z λ λ 2 We can solve the system via a least-squares method (as the one presented for camera calibration) and obtain X λ, from which extract X (first 4 elements of X λ ). The method can be applied to an arbitrary number of cameras: assuming to have N cameras and to know N 2D points x i, projections of the 3D point X that needs to be reconstructed, we get equation (4.), with A =. ], P x P 2 0 x P x P N x N, and X λ = X λ λ 2 λ 3... λ N Triangulation A more robust method is shown in [3], and can be stated as follows. We consider the reconstruction achieved by two cameras, and then generalize to N cameras. From (2.6) applied to cameras C and C 2, we have λ x = P X λ 2 x 2 = P 2 X (4.2) We can re-write (4.2) as x P X = 0 x 2 P 2 X = 0 (4.3) 6

18 where denotes the product vector. Moreover, the first equation of (4.3) can be written as x p T y X p T 2 X = 0, (4.4) p T 3 X where p it, i =, 2, 3 denotes the rows of matrix P. The same can be done for P 2. Taking the product vector for the first equation of (4.3), we get y p T 3 X p T p T X x p T 2 X 3 X X x p T 2 X y p T = 0, (4.5) being the generic product vector a b, a = [a, a 2, a 3 ] T, b = [b, b 2, b 3 ] T, equals to [a 2 b 3 a 3 b 2, a 3 b a b 3, a b 2 a 2 b ] T. Ignoring the third equation (being a linear combination of the other two), and repeating the same procedure for camera C 2, we can arrange the equations to have y p T 3 p T 2 p T x p T 3 y 2 p 2T 3 p 2T 2 p 2T x 2 p 2T 3 X = AX = 0, (4.6) Using the same procedure detailed for camera calibration, by taking the singular value decomposition of A and using the last column of V (normalizing with the last value of V ), we get the solution X. The same procedure can be applied using N cameras C,.., C N. Each camera adds two rows to equation (4.6), and we get y p T 3 p T 2 p T x p T 3 y 2 p 2T 3 p 2T 2 p 2T x 2 p 2T 3... y N p N T 3 p N T 2 p N T x N p N T 3 X = AX = 0, (4.7) This homogeneous equation system can be resolved in the same way as equation (4.6) has been resolved D world point recostruction Assuming that homographies H i are known for each camera i, we can localize a generic 2D point X = (X, Y, ) T using camera matrices and corresponding pixels x i = (x i, y, ) T on each camera, that are the projection of the same X onto the two image planes. 7

19 4.2. Intersection The camera intersection [4] method can be set up also for homographies, and can be stated as follows. From (2.) applied to both cameras, we have λ x = H X λ 2 x 2 = H 2 X. This pair of equations can be rearranged in matrix form as with and A = AX λ = 0, (4.8) [ ] H x 0 H 2 0 x 2, X λ = X Y λ λ 2 We can solve the system via a least-squares method and obtain X λ, from which extract X (first 3 elements of X λ ). The method can be applied to an arbitrary number of cameras: assuming to have N cameras and to know N 2D points x i, projections of the 2D world point X that needs to be reconstructed, we get equation (4.8), with A =. H x H 2 0 x H x H N x N, and X λ = X λ λ 2 λ 3... λ N. 8

20 4.2.2 Triangulation We consider camera triangulation for homographies. cameras C and C 2, we have λ x = H X λ 2 x 2 = H 2 X From (2.) applied to (4.9) We can re-write (4.9) as x H X = 0 x 2 H 2 X = 0 (4.0) where denotes the product vector. Moreover, the first equation of (4.0) can be written as x h T y X h T 2 X = 0, (4.) h T 3 X where h it, i =, 2, 3 denotes the rows of matrix H. The same can be done for H 2. Taking the product vector for the first equation of (4.0), we get y h T 3 X h T h T X x h T 2 X 3 X X x h T 2 X y h T = 0, (4.2) Ignoring the third equation (being a linear combination of the other two), and repeating the same procedure for camera C 2, we can arrange the equations to have y h T 3 h T 2 h T x h T 3 y 2 h 2T 3 h 2T 2 h 2T x 2 h 2T 3 X = AX = 0, (4.3) Using the same procedure detailed for camera calibration, by taking the singular value decomposition of A and using the last column of V (normalizing with the last value of V ), we get the solution X. The same procedure can be applied using N cameras C,.., C N. Each camera adds two rows to equation (4.3), and we get y h T 3 h T 2 h T x h T 3 y 2 h 2T 3 h 2T 2 h 2T x 2 h 2T 3... y N h N T 3 h N T 2 h N T x N h N T 3 9 X = AX = 0, (4.4)

21 This homogeneous equation system can be resolved with the same procedure defined for (4.3). 20

22 Bibliography [] Agnieszka Bier and Leszek Luchowski. Error analysis of stereo calibration and reconstruction. In Andr Gagalowicz and Wilfried Philips, editors, Computer Vision/Computer Graphics CollaborationTechniques, volume 5496 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg, [2] G.H. Golub and C. Reinsch. Singular value decomposition and least squares solutions. Numerische Mathematik, 4(5): , 970. [3] Richard Hartley and Andrew Zisserman. Multiple View Geometry in Computer Vision, 2ed. Cambridge, [4] Gerard Medioni and Sing Bing Kang. Emerging Topics in Computer Vision. Prentice Hall, [5] F.A.N. Palmieri, F. Castaldo, and G. Marino. Harbour surveillance with cameras calibrated with ais data. In Aerospace Conference, 203 IEEE, pages 8, 203. [6] Thodore Papadopoulo and Manolis I. A. Lourakis. Estimating the jacobian of the singular value decomposition: Theory and applications. In David Vernon, editor, ECCV (), volume 842 of Lecture Notes in Computer Science, pages Springer, [7] Emanuele Trucco and Alessandro Verri. Introductory Techniques for 3-D Computer Vision. Prentice Hall,