System and method for quantifying deformation, disruption, and development in a sample

1. A computer-implemented method for determining a first quantification of a deformation on a surface or on incremental regions of a sample, said method implemented using a computing device in communication with a memory, said method comprising:

receiving, by the computing device, a first stereo image and a second stereo image of the sample, each stereo image comprising a camera 1 image and a camera 2 image obtained by first and second cameras, respectively;

registering the first stereo image to the second stereo image, wherein registering the first stereo image to the second stereo image comprises:

using a first warping function W 1 (x;p) to map a plurality of pixels from the camera 1 image of the first stereo image to the camera 1 image of the second stereo image; and

using a second warping function W 2 (x;p) to map a plurality of pixels from the camera 2 image of the first stereo image to the camera 2 image of the second stereo image;

wherein x is a 3D position within a world coordinate system and p is a set of shared warp parameters;

determining a first deformation gradient tensor for the sample based on the first and second warping functions, the first deformation gradient tensor including at least a portion of the first and second warping functions; and

determining the first quantification of the deformation of the sample based at least in part on the first deformation gradient tensor;

wherein the first and second warping functions each comprise an affine warp with a linear translation.

2. The method of claim 1 , further comprising optimizing the first and second warping functions to minimize a difference between each mapped pixel value of the plurality of pixels in the first stereo image and each corresponding pixel value of the plurality of pixels in the second stereo image using an energy minimization method.

3. The method of claim 1 , wherein the first and second warping functions are expressed as:

W

1

⁡

(

x

;

p

)

=

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

⁢

Λ

i

⁢

C

i

⁢

R

1

⁢

⁢

p

⁢

K

2

⁡

[

X

2

Y

2

1

]

+

T

1

⁢

⁢

p

;

and

W

2

⁡

(

x

;

p

)

=

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

⁢

Λ

i

⁢

C

i

⁢

R

1

⁢

⁢

p

T

⁢

R

12

⁢

K

1

⁡

[

X

1

Y

1

1

]

+

T

1

⁢

⁢

p

+

T

12

wherein:

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

is a matrix containing a rotation matrix R i and a translation vector

[

T

i

,

x

T

i

,

y

T

i

,

z

]

describing a translation and rotation from the camera 1 and camera 2 images of the first stereo image to corresponding camera 1 and camera 2 images of the second stereo image;

Λ i is any deformation tensor, scalar, or vector describing a deformation from the camera 1 and camera 2 images of the first stereo image to corresponding camera 1 and camera 2 images of the second stereo image;

C i is a normalization factor relating a normalized coordinate system to a world coordinate system;

R 1p is a 3D rotation matrix describing a rotation from the world coordinate system to a coordinate system of camera 1;

R 12 is a 3D rotation matrix describing a rotation from the coordinate system of camera 1 to a coordinate system of camera 2;

K 1 and K 2 are calibrations for cameras 1 and 2, respectively;

[

X

1

Y

1

1

]

and

[

X

2

Y

2

1

]

are positions within the coordinate systems of cameras 1 and 2, respectively;

T 1p is a 3D translation matrix describing a translation from the world coordinate system to the coordinate system of camera 1; and

T 12 is a 3D translation matrix describing a translation from the coordinate system of camera 1 to the coordinate system of camera 2.

4. The method of claim 1 , further comprising:

determining a displacement field by comparing the first stereo image to the second stereo image;

determining a second deformation gradient tensor for the sample based on a fit or differentiation of the displacement field;

determining a second quantification of the deformation of the sample based at least in part on the second deformation gradient tensor; and

determining a strain concentration tensor comprising the difference between the first quantification of the deformation of the sample and the second quantification of the deformation of the sample.

5. A computing device for determining a first quantification of a deformation on a surface or on incremental regions of a sample, said computing device comprising a processor communicatively coupled to a memory, said computing device programmed to:

receive, by the computing device, a first stereo image and a second stereo image of the sample, each stereo image comprising a camera 1 image and a camera 2 image obtained by first and second cameras, respectively;

register the first stereo image to the second stereo image, wherein registering the first stereo image to the second stereo image comprises:

using a first warping function W 1 (x;p) to map a plurality of pixels from the camera 1 image of the first stereo image to the camera 1 image of the second stereo image; and

using a second warping function W 2 (x;p) to map a plurality of pixels from the camera 2 image of the first stereo image to the camera 2 image of the second stereo image;

wherein x is a 3D position within a world coordinate system and p is a set of shared warp parameters;

determine a first deformation gradient tensor for the sample based on the first and second warping functions, the first deformation gradient tensor including at least a portion of the first and second warping functions; and

determine the first quantification of the deformation of the sample based at least in part on the first deformation gradient tensor;

wherein the first and second warping functions each comprise an affine warp with a linear translation.

6. The computing device of claim 5 , wherein the computing device is further programmed to optimize the first and second warping functions to minimize a difference between each mapped pixel value of the plurality of pixels in the first stereo image and each corresponding pixel value of the plurality of pixels in the second stereo image using an energy minimization method.

7. The computing device of claim 5 , wherein the first and second warping functions are expressed as:

W

1

⁡

(

x

;

p

)

=

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

⁢

Λ

i

⁢

C

i

⁢

R

1

⁢

⁢

p

⁢

K

2

⁡

[

X

2

Y

2

1

]

+

T

1

⁢

⁢

p

;

and

W

2

⁡

(

x

;

p

)

=

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

⁢

Λ

i

⁢

C

i

⁢

R

1

⁢

⁢

p

T

⁢

R

12

⁢

K

1

⁡

[

X

1

Y

1

1

]

+

T

1

⁢

⁢

p

+

T

12

wherein:

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

is a matrix containing a rotation matrix R i and a translation vector

[

T

i

,

x

T

i

,

y

T

i

,

z

]

describing a translation and rotation from the camera 1 and camera 2 images of the first stereo image to corresponding camera 1 and camera 2 images of the second stereo image;

Λ i is any deformation tensor, scalar, or vector describing a deformation from the camera 1 and camera 2 images of the first stereo image to corresponding camera 1 and camera 2 images of the second stereo image;

C i is a normalization factor relating a normalized coordinate system to a world coordinate system;

R 1p is a 3D rotation matrix describing a rotation from the world coordinate system to a coordinate system of camera 1;

R 12 is a 3D rotation matrix describing a rotation from the coordinate system of camera 1 to a coordinate system of camera 2;

K 1 and K 2 are calibrations for cameras 1 and 2, respectively;

[

X

1

Y

1

1

]

and

[

X

2

Y

2

1

]

are positions within the coordinate systems of cameras 1 and 2, respectively;

T 1p is a 3D translation matrix describing a translation from the world coordinate system to the coordinate system of camera 1; and

T 12 is a 3D translation matrix describing a translation from the coordinate system of camera 1 to the coordinate system of camera 2.

8. The computing device of claim 5 , further comprising:

determining a displacement field by comparing the first stereo image to the second stereo image;

determining a second deformation gradient tensor for the sample based on a fit or differentiation of the displacement field;

determining a second quantification of the deformation of the sample based at least in part on the second deformation gradient tensor; and

determining a strain concentration tensor comprising the difference between the first quantification of the deformation of the sample and the second quantification of the deformation of the sample.

9. A system for determining a determining a first quantification of a deformation on a surface or on incremental regions of a sample, said system comprising:

an imaging device communicatively coupled to a computing device; and

the computing device comprising a processor communicatively coupled to a memory, said computing device programmed to:

receive, in the memory, a first stereo image and a second stereo image of the sample, each stereo image comprising a camera 1 image and a camera 2 image obtained by first and second cameras, respectively;

register the first stereo image to the second stereo image, wherein registering the first stereo image to the second stereo image comprises:

using a first warping function W 1 (x;p) to map a plurality of pixels from the camera 1 image of the first stereo image to the camera 1 image of the second stereo image; and

using a second warping function W 2 (x;p) to map a plurality of pixels from the camera 2 image of the first stereo image to the camera 2 image of the second stereo image;

wherein x is a 3D position within a world coordinate system and p is a set of shared warp parameters;

determine a first deformation gradient tensor for the sample based on the first and second warping functions, the first deformation gradient tensor including at least a portion of the first and second warping functions; and

determine the first quantification of the deformation of the sample based at least in part on the first deformation gradient tensor;

wherein the first and second warping functions each comprises an affine warp with a linear translation.

10. The system of claim 9 , wherein the computing device is further programmed to optimize the first and second warping functions to minimize a difference between each mapped pixel value of the plurality of pixels in the first stereo image and each corresponding pixel value of the plurality of pixels in the second stereo image using an energy minimization method.

11. The system of claim 9 , wherein the first and second warping functions are expressed as:

W

1

⁡

(

x

;

p

)

=

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

⁢

Λ

i

⁢

C

i

⁢

R

1

⁢

⁢

p

⁢

K

2

⁡

[

X

2

Y

2

1

]

+

T

1

⁢

⁢

p

;

and

W

2

⁡

(

x

;

p

)

=

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

⁢

Λ

i

⁢

C

i

⁢

R

1

⁢

⁢

p

T

⁢

R

12

⁢

K

1

⁡

[

X

1

Y

1

1

]

+

T

1

⁢

⁢

p

+

T

12

wherein:

[

.

.

.

T

i

,

x

.

R

i

.

T

i

,

y

.

.

.

T

i

,

z

]

is a matrix containing a rotation matrix R i and a translation vector

[

T

i

,

x

T

i

,

y

T

i

,

z

]

describing a translation and rotation from the camera 1 and camera 2 images of the first stereo image to corresponding camera 1 and camera 2 images of the second stereo image;

Λ i is any deformation tensor, scalar, or vector describing a deformation from the camera 1 and camera 2 images of the first stereo image to corresponding camera 1 and camera 2 images of the second stereo image;

C i is a normalization factor relating a normalized coordinate system to a world coordinate system;

R 1p is a 3D rotation matrix describing a rotation from the world coordinate system to a coordinate system of camera 1;

R 12 is a 3D rotation matrix describing a rotation from the coordinate system of camera 1 to a coordinate system of camera 2;

K 1 and K 2 are calibrations for cameras 1 and 2, respectively;

[

X

1

Y

1

1

]

and

[

X

2

Y

2

1

]

are positions within the coordinate systems of cameras 1 and 2, respectively;

T 1p is a 3D translation matrix describing a translation from the world coordinate system to the coordinate system of camera 1; and

T 12 is a 3D translation matrix describing a translation from the coordinate system of camera 1 to the coordinate system of camera 2.

12. The system of claim 9 , further comprising:

determining a displacement field by comparing the first stereo image to the second stereo image;

determining a second deformation gradient tensor for the sample based on a fit or differentiation of the displacement field;

determining a second quantification of the deformation of the sample based at least in part on the second deformation gradient tensor; and

determining a strain concentration tensor comprising the difference between the first quantification of the deformation of the sample and the second quantification of the deformation of the sample.