Cavity, concavityCavity bounded connected component of background (a hollow in an object)Concavity - concave shapes of the contour of an object.2D hole = 2D cavitycavity, 2D holeconcavityconcavity

3D hole and 3D cavityThere is no good precise mathematical definition of a 3D hole.There is only a method how we can detect a holeThe presence of a hole in X is detected whenever there is a closed path in X that cannot be iteratively deformed in X to a single point. [Kong 89]According to the method a hole is not a subset of 3D space.[Coup 10a]

3D hole closing algorithm[Coup 10a]

3D hole closing algorithm[Coup 10a]

3D hole closing algorithm[Coup 10a]

3D hole closing algorithm[Coup 10a]

3D hole closing algorithm (i-th iteration)[Coup 10a]

3D hole closing algorithm (last but one iteration)Surface points T = 1; Tb = 2Curve point (1D isthmus point) T = 2; Tb = 1[Coup 10a]

3D hole closing algorithmSurface points T = 1; Tb = 2[Coup 10a]

3D hole closing algorithm

Let (Z3, m, n, B) is a digital image and Y is a rectangular prism set of black points such that B Y.

The algorithm iteratively removes border points x of Y \ B such that Tb(x) = 1. The removing process is guided by a distance function of point x from set B. [Actouf 2002]

3D hole closing examples[Aktouf 02]

3D hole closing examples[Coup 10a]

3D selective closing of holesInput set: B, Y rectangular prism: B YThe algorithm in addition deletes each point x which is a surface point (Tb(x) = 2 and T(x) = 1) and d(x, B) > a

[Aktouf 02]

Properties of 3D hole closing algorithmLinear in time and space (optimal).Based on well defined topological characteristic of points.No parameters to tune or only one parameter (size of a hole) when it utilises a distance map to close only small holes.wide potential application (reparation of tomografic images, fast detection of holes).

DrawbacksCloses not only holes but also cavities.Sensitive to branches which are close to a hole.

HCA closes holes and cavities

HCA which closes only holes

Branches insensitive AZO

HCA+HCA+ ( Input X, Output Z)01. Xcf CavitiesFilling(X)02. Y HCA(Xcf )03. Ydil GeoDilat(Y;X)04. C Ydil X04. SC UHS(C)05. Z HCA(SC)

HCA+ results

Filtered skeleton constrained by medial axisThe algorithm uses the following notions: quadratic Euclidean distance, medial axis, Euclidean skeleton, bisector function.DenotesLet denote by E the discrete plane Z2, by N the set of nonnegative integers, and by N* the set of strictly positive integers. A point x in E is defined by (x1, x2) with xi in Z. Let x, y E. Let denote by d2(x,y) the square of the Euclidean distance between x and y, that is, d2(x,y) = (x1, y1)2 + (x2 y2)2. Let Y E, let denote by d2(x,Y) the square of the Euclidean distance between x and the set Y, that is, d2(x,Y) = min{d2(x,y); yY}.

Squared Euclidean distance mapLet X be (the object), we denote by the map from E to N which associates, to each point x of E, the value where denotes the complementary of X (the background). The map is calledthe (squared Euclidean) distance map of X.

How to efficiently calculate the squared Euclidean distance

Thickness of an objectLet E = Z2 or E = R2, X E, and let x X. The thickness t(x) of the object X at x is defined as a radius of a biggest ball among balls centred at x and included in X. [Atta 96]

ProjectionLet X be a nonempty subset of E, and let x X. The projection of x on X, denoted by Pr(x,X), is the set of points y of which are at minimal distance from x. [Coup 07]

Bisector functionLet X E, and let x X. The bisector angle of x in X, denoted by X(x), is the maximal unsigned angle between the vectors , , for all y, z in Pr(x, X).In particular, if #Pr(x, X) = 1, then X(x) = 0.The bisector function of X, denoted by X , is the function which associates to each point x of X, its bisector angle in X.

Bisector function in continuous spaceLet E = R2, X E, and let x X. If x belongs to the medial axis of X, then its bisector angle is strictly positive.The bisector angle is equal to for points where the thickness of the object is extremum [Vinc 91].

Bisector function in discreet space

Extended projectionLet E = Z2, X E and x X. The extended projection of x on X, denoted by EPr(x, X), is the union of the sets Pr(y; X), for all y in the 4-neighborhood of x (6-neighborhood in Z3).

Extended bisector functionLet X Z2, and let x X. The extended bisector angle of x in X, denoted by , is the maximal unsigned angle between the vectors , , for all y, z in EPr(x,X).The extended bisector function of X, denoted by , is the function which associates to each point x of X, its extended bisector angle in X.

Niece-looking statementLet X Z2, and let x X. If x belongs to the medial axis of X, then its extended bisector angle is strictly positive.[Coup 07]

Examples of extended bisector functions part 1[Coup 07]

Examples of extended bisector function part 2[Coup 07]

Examples of extended bisector function part 4[Coup 07]

Using a bisector function for medial axis filtering in 2D[Coup 07]

Using extended bisector function for medial axis filtering in 3D Medial axisExtended Bisector Filtered medial axis. Ext. Bis. Threshold: 2.7

Ultimate skeleton constrained by a setLet (E, m, n, B) be a digital image where E = Z2 lub Z3 and B is a finite subset of black points.Def of thinning is presented in slide 19. It is a process of iterative deletion of simple points form an input object B. We say that Y B is an ultimate skeleton of B if Y is an result of thinning of B and there is no simple point for Y.Let C be a subset of B. We say that Y is an ultimate skeleton of B constrained by C if the following conditions are true:C YY is a result of thinning of Bthere is no simple point in B \ C

The set C is called the constraint set relative to this skeleton [Vinc 91].

Ultimate skeleton, constrained by a set and distance-guidedLet C be a subset of B. We say that Y is an ultimate skeleton of B constrained by C and distance-guided if Y is an ultimate skeleton of B constrained by C and the deletion of points is guided by a distance function d in order to select first the points which are closest to the background.

UltimateGuidedSkeleton (Input B, d, C, Output Z)01. Z B02. Q {(d(x), x); where x is any point of B \ Y }03. While Q ; Do04.choose (d(x), x) in Q such that d(x) is minimal05. If x is simple for Z then06. Z Z \ {x}07.Q Q {(d(y), y); where y Nm(x) (Z \ Y)}

Filtered euclidean skeletonFilteredGuidedSkeleton (Input: B, r, , Output: Z)01. DB ExactSquaredEuclideanDistanceMap(B)02. M MedialAxis(X, DB )03. ExactBisector(M, DB )05. Y {x M; DB(x) r and (x) }06. Z UltimateSkeleton(Z, DB, Y )

Example in 2DNon filtered filtered r = 0, = 2 filt. r = 64, = 2.2 filt. R=100, = 3.14 [Coup 07]

Example in 3D Medial axisEuclidean skeletonFiltered skeleton r = 1, = 1.5 [Coup 07]

Example in 3Dr = 15; alfa = 1.5[Coup 07]

References[Aktouf 02] Aktouf Z., Bertrand G., Perroton L.: A three-dimensional holes closing algorithm, Pattern Recognition Letters, vol. 23, pp. 523-31, 2002.[Alex 71] Alexander J. C., Thaler A. I., The boundary count of digital pictures, J Assoc. Comput Mach, pp. 105-112, 1971 [Atta 96] Attali D., Montanvert A., Modeling noise for a better simplification of skeletons, in Proc. of International Conference on Image Processing, 1996, pp. III: 13-16.[Coup 07] Couprie M., Coeurjolly D., Zrour R., Discrete bisector function and Euclidean skeleton in 2D and 3D, Image Vision Comput., vol. 25, pp. 1543-1556, 2007.[Coup 10] Michel Coupries webpage on simple points: http://www.esiee.fr/~info/ck/CK_simple.html[Coup 10a] Michel Coupries presentation on hole closing[Duda 67] Duda R. O., Hart P. E., H. M. J., Graphical-Data-Processing Research Study and Experimental Investigation, AD657670, 1967.[Hild 83] Hilditch C. J., Comparison of thinning algorithms on a parallel processor, Image Vision Comput, pp. 115-132, 1983.[Kong 89] Kong T. Y., A digital fundamental group, Computer Graphics, vol. 13, pp. 159-166, 1989.[Malan 10] George Malandains webpage on digital topology:http://wwwsop.inria.fr/epidaure/personnel/malandain/topology/[Malina 02] Malina W., Ablameyko S., Pawlak W.: Podstawy cyfrowego przetwarzania obrazw, AOW Exit, 2002 (in polish)[Mylo 71] Mylopoulos J., Pavlidis T., On the topological properties of quantized spaces II: Connectivity and order of connectivity, J. Assoc. Comput. Mach, pp. 247-254, 1971.[Rose 70] Rosenfeld A., Connectivity in Digital Pictures, J. ACM, vol. 17, pp. 146-160, 1970.[Rose 73] Rosenfeld A., Ares and curves in digital pictures, J. Assoc. Comput. Mach. 20, 1973, 81 87.[Rose 86] Ronse C., A topological characterization of thinning, Theoret. Comput. Sci. 43, 1986, 31-41.[Stef 71] Stefanelli R., Rosenfeld A., Some parallel thinning algorithms for digital pictures, J. Assoc. Comput. Mach, pp. 255-264, 1971. [Vinc 91] Vincent L.: Efficient computation of varous types of skeletons. In SPIEs Medical Imaging V, volume 1445, San Jose, CA, February 1991.

Czas 0:45There is a hole in 3D volumetric object if there is a closed path inside the object which can not be homotopically transformed into one point which belongs to the object.The presence of a hole in X is detected whenever there is a closed path in X that cannot be iteratively deformed in X to a single point.Czas :1:30

Czas: 0:30Czas: 0:30Pale blueCzas: 0:40Czas: 0:25Czas: 0:15Czas: 1:0Czas 0:25Now let present the algorithm in more formal way.It mens that as a first points which are farthest from B are analised and deletedCzas: 0:45The bottom image shows the result if we do not use a distance function to guide the deletion process. The hole closing path is not centred in the object.Czas: 1:0Czas 1:10We can obtain very interesting effect if we little miWe can obtain interesting effect if we add a little modification to the algo. In addition the algoAs an result the algorithm opens all holes in object whose diameter is bigger that a.Czas 3:0Czas 0;45Czas: 0:48Czas: 3:0On the next several slides I will present a state of the art. Algoritm of skeleton generation for 3D objects.We would like to have similar features for digital space.InputMedial axisEuclidean skeleton Zoomed viev of some details of medial axis and euclidean skeleton filtered skeleton r=15 alfa 1.5