Image model based on n-pixels and defined in algebraic topology, and applications thereof

1 . A computational image model, comprising:

an image support including a structure of n-pixels comprising pixel faces;

quantities related to image features; and

an algebraic structure relating the quantities to the n-pixels and/or pixel faces, the algebraic structure comprising algebraic operations defining a relation between the quantities.

2 . A computational image model as defined in claim 1 , wherein each n-pixel is defined as a geometrical structure comprising vertices, edges, faces and a volume, and wherein each n-pixel comprises:

a first pixel dimension n=0 including the vertices of the n-pixel;

a second pixel dimension n=1 including the edges of the n-pixel;

a third pixel dimension n=2 including the faces of the n-pixel;

a fourth pixel dimension n=3 including the volume of the n-pixel; and

a n th pixel dimension n including the hypervolume of the n-pixel.

3 . A computational image model as defined in claim 1 , wherein the geometrical structure is selected from the group consisting of: a cube, a triangle, a hexagone and a pentagons.

4 . A computational image model as defined in claim 1 , wherein the quantities related to image features are selected from the group consisting of: scalar quantities, vectors, tensors and matrices.

5 . A computational image model as defined in claim 1 , wherein the algebraic operations comprise problem-independent operations.

6 . A computational image model as defined in claim 1 , wherein the algebraic operations comprise problem-dependent operations.

7 . A computational image model as defined in claim 1 , wherein the structure of n-pixels comprises pairs of disjoint n-pixels.

8 . A computational image model as defined in claim 1 , wherein the structure of n-pixels comprises pairs of n-pixels intersecting through a common i-pixel, where i<n.

9 . A computational image model as defined in claim 1 , wherein each n-pixel is translated algebraically into a q-pixel, wherein q ε {1, 2, . . . , n}.

10 . A computational image model as defined in claim 9 , wherein each q-pixel includes (q−1)-faces, (q−2)-faces, . . . , (q-q)-faces.

11 . A computational image model as defined in claim 9 , wherein the image support comprises a geometrical complex, which is a collection of q-pixels.

12 . A computational image model as defined in claim 10 , wherein the image support comprises a geometrical complex, which is a collection of q-pixels, and wherein:

every face of a q-pixel in the geometrical complex is also located in the geometrical complex; and

any pair of two q-pixels of the geometrical complex have an intersection which is either empty or constituted by a common face of both q-pixels of the pair.

13 . A computational image model as defined in claim 11 , comprising a plurality of image supports forming the geometrical complex.

14 . A computational image model as defined in claim 11 , wherein the geometrical complex is expressed in algebraic form as a q-chain, which is a linear combination of all the q-pixels of the geometrical complex.

15 . A computational image model as defined in claim 9 , wherein the geometrical complex comprises q-cochains, which are relations associating quantities related to image features to the q-pixels and/or faces of said q-pixels.

16 . A computational image model as defined in claim 15 , wherein the quantities related to image features and associated to the q-pixels and/or faces of said q-pixels are global quantities associated to all the q-pixels.

17 . A computational image model as defined in claim 15 , wherein the quantities related to image features and associated to the q-pixels and/or faces of said q-pixels are local quantities each associated to one q-pixel and/or faces of said one q-pixel.

18 . A computational image model as defined in claim 16 , comprising (q≧1)-cochains to represent the local quantities.

19 . A computational image model as defined in claim 17 , comprising 0-cochain to represent the global quantities.

20 . A computational image model as defined in claim 17 , wherein the algebraic operations comprise a coboundary operation giving a relationship between the q-cochains.

21 . A computational image model as defined in claim 9 , wherein:

the image support comprises a plurality of geometrical complexes, each being a collection of q-pixels; and

the algebraic operations comprise a codual operation establishing a link between q-cochains that belong to different geometrical complexes.

22 . A method of computationally modelling an image, comprising:

producing an image support including a structure of n-pixels comprising pixel faces;

defining quantities related to image features; and

relating the quantities to the n-pixels and/or pixel faces through an algebraic structure, and relating the quantities to each other through algebraic operations.

23 . A method of computationally modelling an image as defined in claim 22 , wherein relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises translating each n-pixel algebraically into a q-pixel, wherein q ε {1, 2, . . . , n}, wherein each q-pixel includes (q−1)-faces, (q−2)-faces, . . . , (q-q)-faces.

24 . A method of computationally modelling an image as defined in claim 22 , wherein producing an image support comprises forming a geometrical complex, which is a collection of q-pixels, and wherein:

every face of a q-pixel in the geometrical complex is also located in the geometrical complex; and

any pair of two q-pixels of the geometrical complex have an intersection which is either empty or constituted by a common face of both q-pixels of the pair.

25 . A method of computationally modelling an image as defined in claim 24 , wherein producing an image support comprises forming a plurality of image supports forming the geometrical complex.

26 . A method of computationally modelling an image as defined in claim 24 , wherein relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises expressing the geometrical complex in algebraic form as a q-chain, which is a linear combination of all the q-pixels of the geometrical complex.

27 . A method of computationally modelling an image as defined in claim 24 , wherein relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises forming, in the geometrical complex, q-cochains which are relations associating quantities related to image features to the q-pixels and/or faces of said q-pixels.

28 . A method of computationally modelling an image as defined in claim 22 , wherein defining quantities related to image features comprises defining global quantities associated to all the q-pixels.

29 . A method of computationally modelling an image as defined in claim 22 , wherein defining quantities related to image features comprises defining local quantities associated to one q-pixel and/or faces of said one q-pixel.

30 . A method of computationally modelling an image as defined in claim 27 , wherein relating the quantities to each other through algebraic operations comprise producing a coboundary operator giving a relationship between q-cochains.

31 . A method of computationally modelling an image as defined in claim 27 , wherein:

producing an image support comprises forming a plurality of geometrical complexes, each being a collection of q-pixels; and

relating the quantities to each other through algebraic operations comprises producing a codual operation establishing a link between cochains that belong to different geometrical complexes.

32 . An image modelling method as defined in claim 27 , wherein relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises expressing a global quantity associated with all q-pixels through a q-cochain such that, for two adjacent q-pixels c q 1 and c q 2 , the q-cochain F q satisfies the relation F q (λ 1 c q 1 +λ 2 c q 2 )=λ 1 F q (c q 1 )+λ 2 F q (c q 2 ), where λ1 and λ2 are integers.

33 . An image modelling method as defined in claim 22 , wherein:

relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises translating each n-pixel algebraically into a q-pixel, wherein q ε {1, 2, . . . , n}, wherein each q-pixel includes (q−1)-faces, (q−2)-faces, . . . , (q-q)-faces;

producing an image support comprises forming geometrical complexes, each being a collection of q-pixels;

relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises:

expressing each geometrical complex in algebraic form as a q-chain, which is a linear combination of all the q-pixels of the geometrical complex;

forming, in the geometrical complexes, q-cochains which are relations associating quantities related to image features to the q-pixels and/or faces of said q-pixels;

relating the quantities to each other through algebraic operations comprises:

producing a coboundary operator giving a relationship between the q-cochains; and

producing a codual operation establishing a link between q-cochains that belong to different geometrical complexes.

34 . A computational framework for solving a problem using an image computationally modelled by means of the method of claim 33 , comprising:

identifying basic laws associated to the problem;

from the identified basic laws, defining quantities related to the problem;

associating the quantities to respective q-cochains;

associating the basic laws related to the problem to respective coboundary and codual operations; and

resolving the resulting algebraic system.

35 . A computational framework as defined in claim 34 , wherein forming geometrical complexes comprises forming first and second geometrical complexes.

36 . A computational framework as defined in claim 35 , wherein identifying basic laws associated to the problem comprises supporting one basic law through the first geometrical complex.

37 . A computational framework as defined in claim 36 , wherein the problem to be solved is a 2D global differential equation for heat flow in a homogeneous medium, and wherein said one basic law is a heat flow law.

38 . A computational framework as defined in claim 37 , wherein associating the quantities to respective q-cochains comprises representing a global quantity of temperature through a 0-cochain, and associating the heat flow law through a 1-cochain.

39 . A computational framework as defined in claim 35 , wherein identifying basic laws associated to the problem comprises supporting one basic law through the second geometrical complex.

40 . A computational framework as defined in claim 39 , wherein the problem to be solved is a 2D global differential equation for heat flow in a homogeneous medium, and wherein said one basic law is a heat source law.

41 . A computational framework as defined in claim 36 , wherein identifying basic laws associated to the problem comprises supporting a second basic law through the second geometrical complex, and wherein associating the basic laws related to the problem to respective coboundary and codual operations comprises representing a constitutive law linking basic laws from the first and second geometrical complexes by a codual operation.

42 . An image modelling method as defined in claim 22 , wherein:

relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises translating each n-pixel algebraically into a q-pixel, wherein q ε {1, 2, . . . , n}, wherein each q-pixel includes (q−1)-faces, (q−2)-faces, . . . , (q-q)-faces;

producing an image support comprises forming a geometrical complex, which is a collection of q-pixels;

relating the quantities to the n-pixels and/or pixel faces through an algebraic structure comprises:

expressing the geometrical complex in algebraic form as a q-chain, which is a linear combination of all the q-pixels of the geometrical complex;

forming, in the geometrical complex, q-cochains which are relations associating quantities related to image features to the q-pixels and/or faces of said q-pixels;

relating the quantities to each other through algebraic operations comprises:

producing coboundary operations giving a relationship between the q-cochains.

43 . A computational framework for solving a problem using an image computationally modelled by means of the method of claim 42 , comprising:

identifying basic laws associated to the problem;

from the identified basic laws, defining quantities related to the problem;

associating the quantities to respective q-cochains;

associating the basic laws related to the problem to respective coboundary operations; and

resolving the resulting algebraic system.

44 . A computational framework for solving a heat transfer problem, comprising:

producing an image support including a structure of n-pixels, the image support comprising:

q-pixels respectively translating the n-pixel algebraically, wherein q ε {1, 2, . . . , n}, and wherein each q-pixel includes (q−1)-faces, (q−2)-faces, . . . , (q-q)-faces;

geometrical complexes each being a collection of q-pixels;

q-chains respectively expressing the geometrical complexes in algebraic form, each q-chain being a linear combination of all the q-pixels of the geometrical complex;

in the geometrical complexes, q-cochains which are relations associating quantities related to image features to the q-pixels and/or faces of said q-pixels; and

a coboundary defining a relation between q-cochains;

computing a q-cochain T of a first of said geometrical complexes as the location of unknown temperatures;

computing a q-cochain H of the first geometrical complex as a global temperature variation;

finding a q-cochain ε of a second geometrical complex as a global energy variation, as a function of the q-cochain H through a linear transformation;

finding the q-cochain ε as a function of the q-cochain T;

defining a q-cochain G of the first geometrical complex from the q-cochain T through a first coboundary operation, transforming the q-cochain G into a q-cochain Q of the second geometrical complex, and defining, from the q-cochain Q and through a second coboundary operation, a q-cochain D of the second geometrical complex as a global diffusion;

defining a q-cochain S of the second geometrical complex as a global source; and

establishing a relation between the q-cochains ε, D and S.

45 . A computational framework for two-dimensional active contour model, comprising:

producing an image support including a structure of n-pixels, the image support comprising:

q-pixels respectively translating the n-pixel algebraically, wherein q ε {1, 2, . . . , n}, and wherein each q-pixel includes (q−1)-faces, (q−2)-faces, . . . , (q-q)-faces;

geometrical complexes each being a collection of q-pixels;

q-chains respectively expressing the geometrical complexes in algebraic form, each q-chain being a linear combination of all the q-pixels of the geometrical complex;

in the geometrical complexes, q-cochains which are relations associating quantities related to image features to the q-pixels and/or faces of said q-pixels; and

a coboundary defining a relation between q-cochains;

computing a displacement q-cochain D of a first of said geometrical complexes;

computing a strain q-cochain S of a second of said geometrical complexes, comprising:

defining an approximate strain function {tilde over (ε)}(x) as a function of the q-cochain D;

expressing the q-cochain S as a function of the approximate strain function and relative positions of the first and second geometrical complexes; and

computing a force q-cochain F of the second geometrical complex as a coboundary of the strain q-cochain S.