2D image mosaic building 2D3 - Ifremer

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Images of the observed scene Matching points Project Exocet/D page 10/16 Points matches validation Scene geometry estimation Extrinsic parameters estimation Figure 4: Steps of camera self-calibration method 3.3.1. Extraction and matching of points Intrinsic parameters estimation In our application, image sequences are composed of small displacements between two successive images. The extraction and the matching of points are carried out by the KLT algorihthm which is detailed in paragraph 3.1. This algorithm extracts features in the first image and tracks them across the sequence. Despite the complexity of underwater images, a great number of features are positively tracked through the sequence. Moreover, the points are well distributed in the image. 3.3.2. Scene geometry The scene geometry can be represented algebraically by the fundamental matrix F . The fundamental matrix links the coordinates q and q' of a same 3D point Q in two images: 'T q F q = 0 ∀i ∈ [1, n] i i The estimation of the fundamental matrix is based on the Hartley’s normalized 8-point algorithm and uses the points detected and tracked by the KLT algorithm. This algorithm requires a set of at least eight matched points q ↔ q . In order to increase ' the estimation accuracy, a set of about 30 or 50 points can be used. But using more than 8 points leads to the presence of false matches which perturb the estimation of F . So, in compensation, a criterion has been integrated to validate the points and remove false matches. 3.3.3. Points validation The validation of features is carried out by an algorithm based on the RANSAC (RANdom Sample Consensus) algorithm. The selection of point matches is based on the accuracy of the equation representing the scene geometry (see 3.3.2). The equation is computed for all the matched points and allows, estimating the fundamental matrix, to determine the matching error. A list of good features is then constituted to estimate the best fundamental matrix. As a result, only the “best” matches are kept to estimate the final fundamental matrix. Deliverable N° 2D3 Report on image mosaic building DOP/CM/SM/PRAO/06.224 i i Grade : 1.0 27/09/2006
3.3.4. Intrinsic parameters estimation Project Exocet/D page 11/16 There are five intrinsic parameters: focal distance f , scale factors and according to k the image axes u and ku v u0 0 v v , and coordinates of principal point of the image and . The intrinsic parameters estimation is carried out using the Mendonça and Cipolla algorithm [4] applied to a set of five images taken at given intervals from a dense sequence. This algorithm is based on the minimization of a cost function which takes the intrinsic parameters as arguments and the fundamental matrix as parameters. The cost function is: C ( K ) =∑∑ σ − σ n n 1 2 i = 1 w ij j > i ij 2 σ ij ij With: • , , , ) u f α = where αu, αv, u0, v0 correspond respectively to the products of the ( 0 0 v K u v α scale factors according to the axis u and v by the focal length and to the coordinates of the intersection of the optical axis with the image plane, • is the degree of confidence of the fundamental matrix estimation, F wij ij 1 2 • σ > σ are the non-zero singular values of the essential matrix E . ij ij 3.3.5. Extrinsic parameters estimation The extrinsic parameters are composed of rotations and translations of the camera around the three axes (twelve parameters). The extrinsic parameters estimation represents the last step of the self-calibration algorithm. This part is function of intrinsic parameters and of the fundamental matrix F: E = T [] t R = K FK x With: • t : the antisymmetric matric associated to the translation vector t, [] x • R : the rotation matrix, • K : the intrinsic parameters matrix. The algorithm firstly determines the translation t. Afterwards, the rotation matrix is estimated by minimizing: 3 ∑ i= 1 E − R i T [] t xi 2 [] xi Where E and t are the i-th row vectors of matrices E and [ t ] x . 3.4. Experiments A statistical comparative study of this camera self-calibration method is presented in [PES03]. Some studies with simulated data have allowed us to show that some trajectories of the underwater vehicle are more adapted for the intrinsic parameters estimation of the camera. Deliverable N° 2D3 Report on image mosaic building DOP/CM/SM/PRAO/06.224 ij Grade : 1.0 27/09/2006
Page 1 and 2: Underwater Systems Department A.G.
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Page 7 and 8: We can note that J ( x) = I( x −
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