Level Sets Methods in Imaging Science Dr. Corina S. Drapaca [email protected] Pennsylvania State University University Park, PA 16802, USA Level Sets Methods in Imaging Science – p.1/36 Introduction Textbooks • S.Osher, R.Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences vol. 153, 2003. • J.A.Sethian, Level Set Methods and Fast Marching Methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge University Press, 2nd edition, 1999. • Handbook of Mathematical Models in Computer Vision, edited by N.Paragios, Y.Chen, O.Faugeras, Springer, 2006. • Geometric Level Set Methods in Imaging, Vision, and Graphics, edited by S.Osher, N.Paragios, Springer, 2003. Level Sets Methods in Imaging Science – p.2/36 Introduction Level Set Methods • Originally introduced by S.Osher, J.A.Sethian, Journal of Computational Physics 79, pg.12-49, 1988. • Extensively used in • computer vision • CAD (computer-aided design) • optimal design and control • computational geometry • computational fluid mechanics • computational physics (two-phase flows, shocks, solid-fluid coupling, epitaxial growth, etc.) • image processing (restoration, segmentation, registration, deblurring/denoising, reconstruction of surfaces from unorganized data points, etc.) • ”Level set methods add dynamics to implicit surfaces.” (Osher & Fedkiw book) • useful in analyzing and computing the motion of an interface bounding a (multiply connected) open region under a given velocity field. Level Sets Methods in Imaging Science – p.3/36 Introduction Implicit Functions In Rn: • (Sub) domains are n-dimensional. • Interface: the border between two subdomains (or between the inside and outside of a domain). • Interfaces have dimension n − 1 and codimension 1. • Interface representations: • explicit: list of all the interface points using a parametrization of the interface. • implicit: the isocontour of a (implicit) function n φ(~x)= C, ∀~x =(x1,...,xn) ∈ R , C=const. For simplicity, C = 0. • We will consider only the case of simple closed interfaces. A simple (left) and a non-simple (right) closed interface. Level Sets Methods in Imaging Science – p.4/36 Introduction Implicit Functions • Examples: − Implicit function φ(x) = x2 − 1 in R defining the inside region Ω = (−1, 1) and the outside region Ω+ = (−∞, −1) ∪ (1, +∞) as well as the boundary (interface) ∂Ω = {−1, 1} (left). Implicit function φ(x, y) = x2 + y2 − 1 in R2 defining the inside region (unit open disk) − Ω = {(x, y) ∈ R2|x2 + y2 < 1}, the outside region Ω+ = {(x, y) ∈ R2|x2 + y2 > 1} and the interface (unit circle) ∂Ω = {(x, y) ∈ R2|x2 + y2 = 1} (right). Level Sets Methods in Imaging Science – p.5/36 Introduction Implicit Functions • The explicit representations of the two interfaces in our examples are: • in R: a list of points {−1, 1} (dimension 0) • in R2: one possible paramaterization of the unit circle is: x = cos s, y = sin s, 0 ≤ s ≤ 2π • We assume that the interfaces’ parametrizations are such that the interior regions are on the left side as the parameters increase. • The connectivity (ordering) of the points of an interface contains important information about the topology (shape) of the interface. • In addition, a moving interface can change its topology and hence its connectivity. • Active contour: name used in the image processing literature for a moving (dynamic) interface. Level Sets Methods in Imaging Science – p.6/36 Introduction Limitations of the explicit representation of an interface: • it is hard to parametrize arbitrary curves in R2 and hyper-surfaces in Rn, n ≥ 3. • the connectivity of hyper-surfaces in Rn, n ≥ 3 is very difficult to represent; hence explicit interfaces cannot handle easily interfaces with holes and topological changes. Advantages of implicit interfaces: • Going from R2 to Rn, n ≥ 3 is straightforward: • For example, the zero isocontour of the implicit function φ(x, y, z)= x2 + y2 + z2 − 1 is the implicit representation of the unit sphere, the interface between the inside open unit ball Ω− = {(x, y, z) ∈ R3|φ(x, y, z) < 0} and its exterior open region Ω+. • Topological changes and discontinuous interfaces are handled naturally, there are no problems with the connectivity of implicit interfaces. • Implicit functions make Boolean operations, geometry and calculus operations easy to apply and use. Level Sets Methods in Imaging Science – p.7/36 Introduction Boolean operations: If φ1 and φ2 are two implicit functions, then the implicit representation of: • the union of the interior regions of φ1 and φ2 is φ = min(φ1,φ2). • the intersectionof the interior regions of φ1 and φ2 is φ = max(φ1,φ2). • the complement of the interior region of φ1 is φ = −φ1. Level Sets Methods in Imaging Science – p.8/36 Introduction Boolean operations: If φ1 and φ2 are two implicit functions, then the implicit representation of: • the union of the interior regions of φ1 and φ2 is φ = min(φ1,φ2). • the intersectionof the interior regions of φ1 and φ2 is φ = max(φ1,φ2). • the complement of the interior region of φ1 is φ = −φ1. Geometry: If φ is a smooth enough implicit function then: • gradient of the interface: ∂φ ∂φ ∂φ ∇φ(~x) = , ,.., , ~x =(x , x ,..,x ) ∈ Rn ∂x ∂x ∂x 1 2 n 1 2 n ∇φ • unit outward normal to the interface: N~ = |∇φ| ∇φ • mean curvature of the interface: κ = ∇· N~ = ∇· |∇φ| Level Sets Methods in Imaging Science – p.8/36 Introduction Boolean operations: If φ1 and φ2 are two implicit functions, then the implicit representation of: • the union of the interior regions of φ1 and φ2 is φ = min(φ1,φ2). • the intersectionof the interior regions of φ1 and φ2 is φ = max(φ1,φ2). • the complement of the interior region of φ1 is φ = −φ1. Geometry: Convex regions have κ > 0, and concave regions have κ < 0. Level Sets Methods in Imaging Science – p.8/36 Introduction Calculus operations: • Characteristic functions χ− and χ+ of the interior region Ω− and, respectively, exterior region Ω+ are: 1, if φ(~x) ≤ 0(i.e. ~x ∈ Ω−) χ−(~x)= ( 0, if φ(~x) > 0 and + 0, if φ(~x) ≤ 0 χ (~x)= + ( 1, if φ(~x) > 0(i.e. ~x ∈ Ω ) • Since χ± are functions of a multidimensional variable ~x, it is easier to replace these characteristic functions with the one-dimensional Heaviside function 0, if φ ≤ 0 H(φ)= ( 1, if φ> 0 Level Sets Methods in Imaging Science – p.9/36 Introduction Calculus operations: • Volume (area, length) integrals of a function f are: f(~x)χ−(~x)d~x = f(~x)(1 − H(φ(~x)))d~x, Rn Rn Z Z and f(~x)χ+(~x)d~x = f(~x)H(φ(~x))d~x, Rn Rn Z Z representing the integrals of f over Ω− and, respectively, Ω+. • Dirac delta distribution: the directional derivative of the Heaviside function H in the normal direction of the normal N~ : ∇φ(~x) δˆ(~x) = ∇H(φ(~x)) · N~ = H′(φ)∇φ(~x) · |∇φ(~x)| = H′(φ)|∇φ(~x)| = δ(φ)|∇φ(~x)| ′ since ∇φ(~x) · ∇φ(~x) = |∇φ(~x)|2 and δ(φ) = H (φ) (the one-dimensional Dirac delta distribution). Level Sets Methods in Imaging Science – p.10/36 Introduction Calculus operations: • Surface (line or point) integral of a function f over the boundary (interface) ∂Ω between the inside Ω− and outside Ω+ regions is: fds = f(~x)δˆ(~x)d~x = f(~x)δ(φ)|∇φ(~x)|d~x. Rn Rn Z∂Ω Z Z • By embedding the volume and surface integrals in higher dimensions, the above formulas avoid the need for identifying inside, outside, or boundary regions, making the numerical integration easier. • The numerical approximations of the above integrals require that the Heaviside function H and the Dirac distribution δ be smooth. Level Sets Methods in Imaging Science – p.11/36 Introduction Calculus operations: • Possible regularizations of H and δ: • C2 regularization of H 0, if φ ≤−ǫ 1 φ 1 πφ H (φ)= 1+ + sin , if |φ|≤ ǫ 2,ǫ 2 ǫ ǫ ǫ 1, if φ>ǫ , φ >ǫ dH (φ) 0 if | | δ (φ)= 2,ǫ = 1 1 1 πφ 2,ǫ dφ + cos , if |φ|≤ ǫ 2 ǫ ǫ ǫ • ∞ C regularization of H 1 2 φ dH (φ) 1 ǫ H (φ)= 1+ arctan ,δ (φ)= ∞,ǫ = ∞,ǫ 2 π ǫ ∞,ǫ dφ π ǫ2 + φ2 • As ǫ → 0, H2,ǫ, H∞,ǫ → H and δ2,ǫ, δ∞,ǫ → δ. Level Sets Methods in Imaging Science – p.12/36 Introduction Level Set Functions • What is a good choice of an implicit function φ? • Signed distance functions: a subset of the implicit functions φ that are positive on the exterior region, negative in the interior region, and zero on the interface. • Examples of signed Euclidean distance functions: • for the interface {−1, 1} in R: φ(x)= |x|− 1 has |∇φ| = 1, ∀x 6= 0. • for the unit circle {(x, y) ∈ R2|x2 + y2 = 1} in R2: φ(x, y)= x2 + y2 − 1 has |∇φ| = 1, ∀~x 6= ~0. • R3 2 2 2 R3 for the unitp sphere {(x, y, z) ∈ |x + y + z = 1} in : φ(x, y, z)= x2 + y2 + z2 − 1 has |∇φ| = 1, ∀~x 6= ~0. • All signed Euclideanp distance functions φ satisfy |∇φ| = 1 almost everywhere, so N~ = ∇φ, κ = ∆φ. • However, avoid using the above expressions for N~ and κ since |∇φ| 6= 1 when using numerical approximations. Level Sets Methods in Imaging Science – p.13/36 Introduction Motivational Example of the Level Set Method in 2D: Sea Level = interface; Oceans = interior region; Mountains = exterior region; φ(x, y, τ) = 0 moving interface; φ(x,y, 0) = signed Euclidean distance z y y x x a. Initial Circle b. Initial Surface y z y x x c. Circle at time d.