Image processing method based on the finite element method for directly solving inverse problems in structural mechanics

1. A method for image processing for formation of an elasticity image of a body, the method comprising the steps of:

receiving a deformation image illustrating a displacement or deformation field of the points of the body as a function of a pressure difference in the body, wherein the deformation image is a deformation elastogram produced by introducing an ultrasonic wave into the body;

defining a meshing of the deformation image, the meshing step applying a finite element method, the meshing including a plurality of elementary cells each delimiting a same material and each including at least three nodes, each node belonging to one or several adjacent cells to the meshing;

assigning to each node i of the meshing a pair of unknown nodal variables, each pair being representative of elastic properties to be determined;

detecting at least one node common to at least two adjacent elementary cells delimiting different materials;

enriching the at least one detected common node the enrichment step including replacing the pair of unknown variables assigned to the detected common node with at least two pairs of enrichment unknown variables, each pair of enrichment unknown variables of the detected common node being assigned to a respective elementary cell from among the at least two adjacent elementary cells delimiting different materials;

calculating an elementary matrix for each elementary cell of the meshing so as to obtain a plurality of elementary matrices, the elementary matrices being calculated by taking into account pairs of unknown variables assigned to the elementary cells, and pairs of unknown variables assigned to the elementary cells;

assembling the elementary matrices of the plurality of elementary matrices in order to obtain a structure matrix;

calculating the elasticity image of the body from the deformation image and from the structure matrix; and

displaying the elasticity image.

2. The method according to claim 1 , further comprising, prior to the step of defining a meshing, a step consisting of segmenting the deformation image in order to obtain a segmented deformation image including a plurality of representative regions of areas of the body assumed to be in different materials.

3. The method according to claim 2 , wherein the step of defining a meshing is applied on the segmented deformation image, the dimensions and positions of the cells of the meshing being determined depending on the positions and dimensions of the regions of the segmented deformation image.

4. The method according to claim 1 , wherein the step of calculating an elasticity image comprises the resolution of the following matrix equation:

{ R }=([ Q′] T [Q ′]) −1 [Q′] T {F′}

wherein:

[Q′] represents the reduced structure matrix, and [Q′] T its transposed,

{F′} represents the reduced matrix of the field of forces applied to the nodes, and

{R} represents the matrix of the field of elasticities of the body at the nodes.

5. The method according to claim 1 , wherein the step of enriching a detected common node comprises the replacement of the pair of unknown variables assigned to the common node detected by n pairs of enrichment unknown variables, n corresponding to the number of different materials at the boundary of which is located the common node.

6. A system for image processing for formation of an elasticity image of a body, the system comprising:

a receiver for receiving a deformation image illustrating a displacement field of the points of the body as a function of a pressure difference in the body, wherein the deformation image is a deformation elastogram produced by introducing an ultrasonic wave into the body;

a processor configured to:

define a meshing of the deformation image by applying a finite element method, the meshing consisting of a plurality of elementary cells each delimiting a same material and each including at least three nodes, each node belonging to one or several cells adjacent to the meshing;

assign to each node i of the meshing a pair of unknown nodal variables ((λ i , μ i ) or (E 1 , ν i )), each pair being representative of elastic properties to be determined;

detect at least one node common to at least two adjacent elementary cells delimiting different materials;

enrich the and at least one detected common node, the enrichment including replacing the pair of unknown variables assigned to the detected common node with at least two pairs of unknown enrichment variables, each pairs of unknown enrichment variables of the detected common node being assigned to a respective elementary cell from among the at least two adjacent elementary cells delimiting different materials;

calculate an elementary matrix for each elementary cell of the meshing so as to obtain a plurality of elementary matrices, the elementary matrices being calculated while taking into account pairs of unknown variables assigned to the elementary cells, and pairs of unknown variables assigned to the elementary cells;

assemble the elementary matrices from the plurality of elementary matrices for obtaining a structure matrix; and

calculate the elasticity image of the body from the deformation image and from the structure matrix; and

a display device,

wherein the processor is further configured to display the elasticity image on the display device.

7. The system according to claim 6 , wherein the processor is configured to segment the deformation image in order to obtain a segmented deformation image consisting of a plurality of representative regions of areas of the body assumed to be in different materials.

8. A computer program product for executing image processing for formation of an elasticity image of a body comprising a non-transitory computer readable medium storing instructions that, when executed by a computer, cause the computer to perform operations comprising:

receiving a deformation image illustrating a displacement or deformation field of the points of the body as a function of a pressure difference in the body, wherein the deformation image is a deformation elastogram produced by introducing an ultrasonic wave into the body;

defining a meshing of the deformation image, the meshing step applying a finite element method, the meshing including a plurality of elementary cells each delimiting a same material and each including at least three nodes, each node belonging to one or several adjacent cells to the meshing;

assigning to each node i of the meshing a pair of unknown nodal variables, each pair being representative of elastic properties to be determined;

detecting at least one node common to at least two adjacent elementary cells delimiting different materials;

enriching the at least one detected common node, the enrichment step including replacing the pair of unknown variables assigned to the detected common node with at least two pairs of enrichment unknown variables, each pair of enrichment unknown variables of the detected common node being assigned to a respective elementary cell from among the at least two adjacent elementary cells delimiting different materials;

calculating an elementary matrix for each elementary cell of the meshing so as to obtain a plurality of elementary matrices, the elementary matrices being calculated by taking into account pairs of unknown variables assigned to the elementary cells, and pairs of unknown variables assigned to the elementary cells;

assembling the elementary matrices of the plurality of elementary matrices in order to obtain a structure matrix; and

calculating the elasticity image of the body from the deformation image and from the structure matrix; and

displaying the elasticity image.