Efficient energies and algorithms for parametric snakes - EPFL

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20whereh 1 (l, m, n) =h 2 (l) =∫ ∞−∞∫ ∞−∞ϕ ′ (t) ϕ ′ (t + l) ϕ ′ (t + m) ϕ ′ (t + n) dt (39)ϕ ′ (t) ϕ ′ (t + l) dt. (40)Note that the multidimensional filtering is performed assuming periodic boundary conditions. Thecomputational complexity is small, since the sum depends only on the coefficient sequence whosenumber is typically much lesser than the number of curve samples. The computational complexityin evaluating the above sum is N 3 M. The filter coefficients (39) and (40) are precomputed asshown in Appendix-C.B. Partial derivatives of the constraint energyComputing the partial derivatives of (28), in all its generality, would give a very complicatedexpression. To make the problem more tractable and to reduce its computational complexity, wemake the assumption that the optimal parameters t i ; i = 0 . . . N c − 1 are known. In this case,(28) gets simplified toE c =and its partial derivatives are given by:⎡ ⎤⎣ ∂E c/∂c x,k⎦ =∂E c /∂c y,kN∑c−1i=0N∑c−1i=0|r (t i ) − r c,i | 2 (41)⎛⎡⎤ ⎡⎝⎣ x c,i⎦ −y c,i⎣ x (t i)y (t i )⎤⎞⎦⎠ ϕ (t i − k) . (42)Using the finite support of the scaling functions, we limit the sum to the relevant indices (we needto evaluate it only for {i| 0 < (t i − k) < N c }). In practice, we resort to a two-step optimizationwhere the snake is first evolved using the above formulas for the derivatives with the current setof t i ’s. The optimal parameters t i are then re-estimated within the loop as:t i = arg mint∈[0,M] |r (t) − r c,i| ; i = 0 . . . N c − 1 (43)C. Estimation of the probability distribution functionsThe evaluation of (14) requires the specification of the probability distribution functionsP (f (s) |s ∈ R) and P (f (s) |s ∈ R). If we do not have any a priori knowledge of thesedistributions, these are estimated iteratively from the image data itself assuming R = S andSubmitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT
21R ′ = S ′ . Note that these assumptions are valid if the snake is close to the real boundary. We usedensities such as the Gaussian distribution which are represented by a few parameters (mean andvariance). The estimation of these parameters require integrating the image and its square in theregion bounded by S. We compute the integrals efficiently using (31) with the correspondingintegrated functions (similar to (33)) precomputed.The estimation of the distributions are followed by a non-linear transformation which mapsf into f u . Since this transformation is time consuming, the estimation of the distributions andthe updating of f u is only performed periodically, typically once every 10 iterations.D. Computation of the length and areaThe computation of the internal energy requires the estimation of the current length of thecurve. We compute the length as a discrete approximation of the integral ∫ M(0 x ′ (t) 2 + y ′ (t) 2) 12dtasLength = 1 RMR−1∑i=0√x ′ ( iR) 2+ y ′ ( iR) 2. (44)The area of the curve is obtained by Green’s theorem as ∮ ydx, which when expanded givesCArea =M−1∑N−1∑k=0 l=−N+1c y,k c p x,lq (k − l) , (45)where q (m) = ∫ ∞−∞ ϕ (t) ϕ′ (t − m) dt is obtained as in [38]. Note that the area obtained bythe above expression is signed; its sign is utilized to determine the direction (clockwise oranti-clockwise) of the curve.VII. EVOLVING THE CURVEA. Optimization AlgorithmAs mentioned before, the active contour algorithm extracts the final contour by finding theminimum of the energy function. Having obtained the partial derivatives, we can use an efficientoptimization algorithm to evolve the contour. Here, we implemented the conjugate gradientand steepest descend algorithms. The conjugate gradient algorithm resulted in slightly fasterconvergence, but was less flexible for loop recovery and knot addition/deletion discussed later.Hence in our final implementation we reverted to the simpler steepest descend algorithm, whichwas found to be entirely satisfactory for our purpose.Submitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT
Page 1 and 2: 1Efficient energies and algorithms
Page 3 and 4: 3• Parametric snakes, where the c
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Page 7 and 8: 71) Gradient magnitude energy: Many
Page 9 and 10: 9(a) Initialization(b) Magnitude-ba
Page 11 and 12: 11The use of this energy assumes tw
Page 14 and 15: 14second term reduces to∫ Mprovid
Page 16 and 17: 16V. EXTERNAL CONSTRAINT ENERGY.As
Page 18 and 19: 18computed as the curve integral∫
Page 22: d22d sOCFig. 6.Computation of the e
Page 25 and 26: 25(a)(b)(c)(d)Fig. 8. Segmentation
Page 27 and 28: 27In the second step, we make use o
Page 29 and 30: 29APPENDIX - D: INTEGRAL OF THE TAN
Page 31: 31[32] L.D. Cohen and I. Cohen, “

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