Efficient energies and algorithms for parametric snakes - EPFL

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10the right is over C. Using the vector notation, we rewrite (12) as∫∮(∇ · F) ds = − k · (F × dr) , (13)SCwhere k is the unit normal to the two dimensional space. Using this identity, we simplify (9) tothe form (10).Note that in the special case when e f = ∇f, we get T e (f) = ∇ 2 f. This means that our newgradient-based energy (7) is equivalent to integrating the Laplacian of the image in the regionbounded by the curve.(a) initialization (b) edge only (α = 1)(c) region only (α = 0) (d) unified (α = 0.5)Fig. 4. Illustration of the use of the unified image energy in the segmentation of a corpus-callossum image (b) The use ofthe gradient based energy fails to converge in regions where the gradient information is absent (c) The region-based energy ismisled by the lack of image contrast (d) The unified energy leads to a good segmentation.B. Region-based image energyRecent research in active contours is increasingly focusing on the use of statistical regionbasedimage energy [11], [19], [24], [25]. This type of energy can provide the snake with vitalboundary information, especially while it is far away from the real contour, thus resulting in alarger basin of attraction.Submitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT
11The use of this energy assumes two main regions in image 4 , with different probability distributions.A simple example is the case where we have to segment a white object from a darkbackground; the regions will have different means and possibly different variances. We use thestatistical formulation of Staib et. al. [10] to specify the region likelihood function:∫E region = − log (P (f (s) |s ∈ R)) dsS∫− log (P (f (s) |s ∈ R ′ )) ds, (14)S ′where R and R ′ denote the different image regions. We denote the regions in the curve andoutside by S and S ′ respectively. It is easy to see that (14) attains a maximum when R = Sand R ′ = S ′ . We rewrite the above integral as∫E region = − log (P (f (s) |s ∈ R)) dsS∫−C + log (P (f (s) |s ∈ R ′ )) ds, (15)Swhere C = ∫ S ′ ⋃ S log (P (f (s) |s ∈ R′ )) ds. Since C does not depend on the position of thecurve and hence we remove it from the cost function. Thus, the region-based cost function issimplified to∫E region =We now give a few examples to illustrate (16).S( )P (f (s) |s ∈ R)− logP (f (s) |s ∈ R ′ )} {{ }T r(f)ds. (16)1) The regions R and R ′ have Gaussian distributions with the same variance. In this case,we obtainwhere µ R > µ R ′T r (f) = − 2 (µ R − µ R ′)σ 2⎛⎜⎝ f − µ R + µ R ′} {{ 2 }µ R,R ′⎞⎟⎠ , (17)are the means of the regions R and R ′ and and σ the standard deviation.The regions of f with values above µ R,R ′are mapped to negative values while the onesbelow are assigned positive values. Hence, evolving the contour using (16) will result inthe curve adjusting itself to have regions of f above µ R,R ′inside while excluding the ones4 This can be generalized to n > 2 regions.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
Page 5 and 6: 5(6,8); t=2C¡¡¡¡¡¡¡¡¡¡¡
Page 7 and 8: 71) Gradient magnitude energy: Many
Page 9: 9(a) Initialization(b) Magnitude-ba
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 20 and 21: 20whereh 1 (l, m, n) =h 2 (l) =∫
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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