Efficient energies and algorithms for parametric snakes - EPFL

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6B. Active contour models: formulationAn active contour, as introduced by Kass et. al. [3], is a curve described as an orderedcollection of points which evolves from its initial position to some boundary within the image.The curve evolution is formulated as an energy minimization; the snake energy is typically alinear combination of three terms:1) the image energy, which is responsible for guiding the snake towards the boundary ofinterest.2) the internal energy, which ensures that the segmented region has smooth boundaries.3) the constraint energy, which provides a means for the user to interact with the snake.The total energy of the snake is written asE snake (Θ) = E image (Θ) + E int (Θ) + E c (Θ) , (4)where Θ is the collection of curve coefficients Θ = (c 0 , c 1 , . . . , c M−1 ). The optimal curveparameters are obtained asΘ = arg minΘ E snake (Θ) . (5)It is obvious that the quality of segmentation is dependent on the choice of the energy terms.We deal with them in detail in the following sections and they are listed in Table. I for easyreference. The energy is minimized iteratively by updating the snake coefficients.III. IMAGE ENERGYThe image energy is the most important of the three energy terms. In this section, we identifysome limitations of the widely-used gradient magnitude energy and propose a new cost functionthat overcomes these problems. We also present a unified framework which includes the edgebasedand region-based approaches as particular cases.A. Edge-based image energyTraditional snakes rely on edge maps derived from the image to be guided to the actualcontour. The most popular approach is based on the magnitude of the gradient.Submitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT
71) Gradient magnitude energy: Many of the parametric snakes described in the literature usethe integral of the square of the gradient magnitude along the curve as the image energy [6],[9], [10], [17]. Mathematically, we haveE mag = −∫ M0|∇f (t)| 2 dt, (6)where ∇f (t) denotes the gradient of f at the point r (t). As pointed out in [22], one disadvantageof this measure is that it does not use the direction of the gradient. At the boundary, the imagegradient is perpendicular to the contour. This extra information can be incorporated into theexternal energy to make it more robust.A more fundamental problem is the dependence of E mag on the parametrization; we obtaina different value of E mag if the curve is represented in terms of a parameter t ′ = w (t), wherew is a monotonically increasing one to one warping function. The use of such an energy maytherefore result in the curve re-adjusting its parametrization in trying to minimize E mag (e.g. withB-spline curves, the knots will move to regions of the contour where the gradient magnitude isrelatively high). This problem is demonstrated in Fig. 3-b.2) New gradient-based image energy: The gradient magnitude energy is the integral of ascalar field derived from the gradient vector field. We propose a new energy that uses the vectorfield directly:∮E grad = −∮= −CCk · (∇f (r) × dr) (7)∇f (r) · (dr × k) , (8)} {{ }||dr|| ˆn(r)where k is the unit vector orthogonal to the image plane. Here ˆn (r) denotes the unit normalto the curve at r. Note that this approach of accounting for the gradient direction is similar inphilosophy to [22], even though the expression used by these authors is different and parameterdependent.This integration process is illustrated in Fig. 2; with our convention, the vector ˆn (r) is theinward unit normal to the curve 3 meaning that we are integrating the component of the gradientorthogonal to the curve. Note that (7) is independent of the parameter t, and hence does not3 The vector k is chosen depending on the direction in which the curve is described, such that ˆn (r) = dr×k||dr||unit normal.is the inwardSubmitted 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 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 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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