Velocity estimation from imagery using symmetric displaced frame difference equation

1. A method for processing an image sequence, comprising;

providing a symmetric displaced frame difference equation;

receiving an input image sequence comprising a pair of image frames individually including multidimensional image data corresponding to a plurality of pixel locations at different times;

using at least one processor, solving the symmetric displaced frame difference equation using:

an iteration equation and the image data of the pair of image frames to determine a displacement field describing displacement vectors at half of the total displacement vector, and

at least one bilinear polynomial function expressing a multidimensional displacement field; and

determining, using the at least one processor, displacement by motion for the input image sequence based on the solved symmetric displaced frame difference equation.

2. The method of claim 1 ,

wherein the multidimensional image data of the pair of frames is two-dimensional, and

wherein the symmetric displaced frame difference equation is

I ( i+u ij Δt h ,j+v ij Δt h ,t 2 )− I ( i−u ij Δt h ,j−v ij Δt h ,t 1 )=0,

wherein:

I is an intensity of the image data,

i is a pixel index in a horizontal direction x,

j is a pixel index in a vertical direction y orthogonal to the horizontal direction x,

u ij is a component of the velocity vector on a pixel point,

v ij is another component of the velocity vector on a pixel point, and

Δt h is equal to Δt/2, wherein Δt is a time difference between the pair of image frames (t 1 and t 2 ).

3. The method of claim 1 , wherein solving the symmetric displaced frame difference equation comprises using at least one iteration equation derived from the symmetric displaced frame difference equation by conversion to a fully or over-constrained system using a nonlinear least squares model of the displacement field.

4. A motion estimator apparatus, comprising:

at least one processor; and

a memory storing a symmetric displaced frame difference equation,

wherein the at least one processor is operative to:

receive an input image sequence comprising a pair of image frames individually including multidimensional image data corresponding to a plurality of pixel locations at different times,

solve the symmetric displaced frame difference equation using;

an iteration equation and the image data of the pair of image frames to determine a displacement field describing displacement vectors at half of the total displacement vector, and

at least one bilinear polynomial function expressing a multidimensional displacement field, and

determine displacement by motion for the input image sequence based on the solved symmetric displaced frame difference equation.

5. The motion estimator apparatus of claim 4 ,

wherein the multidimensional image data of the pair of frames is two-dimensional, and

wherein the symmetric displaced frame difference equation is

I ( i+u ij Δt h ,j+v ij Δt h ,t 2 )− I ( i−u ij Δt h ,j−v ij Δt h ,t 1 )=0,

wherein:

I is an intensity of the image data,

i is a pixel index in a horizontal direction x,

j is a pixel index in a vertical direction y orthogonal to the horizontal direction x,

u ij is a component of the velocity vector on a pixel point,

v ij is another component of the velocity vector on a pixel point, and

Δt h is equal to Δt/2, wherein Δt is a time difference between the pair of image frames (t 1 and t 2 ).

6. The motion estimator apparatus of claim 4 , wherein the at least one processor is operative to solve the symmetric displaced frame difference equation using at least one iteration equation derived from the symmetric displaced frame difference equation by conversion to a fully or over-constrained system using a nonlinear least squares model of the displacement field.

7. A non-transitory computer readable medium with computer executable instructions for:

providing a symmetric displaced frame difference equation;

receiving an input image sequence comprising a pair of image frames individually including multidimensional image data corresponding to a plurality of pixel locations at different times;

solving the symmetric displaced frame difference equation using:

an iteration equation and the image data of the pair of image frames to determine a displacement field describing displacement vectors at half of the total displacement vector, and

at least one bilinear polynomial function expressing a multidimensional displacement field; and

determining displacement by motion for the input image sequence, based on the solved symmetric displaced frame difference equation.

8. The non-transitory computer readable medium of claim 7 :

wherein the multidimensional image data of the pair of frames is two-dimensional, and

wherein the symmetric displaced frame difference equation is

I ( i+u ij Δt h ,j+v ij Δt h ,t 2 )− I ( i−u ij Δt h ,j−v ij Δt h ,t 1 )=0,

wherein:

I is an intensity of the image data,

i is a pixel index in a horizontal direction x,

j is a pixel index in a vertical direction y orthogonal to the horizontal direction x,

u ij is a component of the velocity vector on a pixel point,

v ij is another component of the velocity vector on a pixel point, and

Δt h is equal to Δt/2, wherein Δt is a time difference between the pair of image frames (t 1 and t 2 ).

9. The non-transitory computer readable medium of claim 7 , comprising computer-executable instructions for solving the symmetric displaced frame difference equation using at least one iteration equation derived from the symmetric displaced frame difference equation by conversion to a fully or over-constrained system using a nonlinear least squares model of the displacement field.

10. The method of claim 1 , wherein receiving the input image sequence comprises receiving only two image frames.

11. The motion estimator apparatus of claim 4 , wherein the input image sequence comprises only two image frames.

12. The non-transitory computer readable medium of claim 7 , wherein receiving the input image sequence comprises receiving only two image frames.

13. The method of claim 1 , further comprising:

determining, using the at least one processor, displacement by motion for the input image sequence based on the solved symmetric displaced frame difference equation.

14. The method of claim 1 , wherein solving the symmetric displaced frame difference equation using the iteration equation and the image data of the pair of image frames to determine a displacement field describing displacement vectors comprises:

determining, using the at least one processor, a displacement vector corresponding to a moving particle in the image sequence.

15. The method of claim 14 , further comprising:

determining, using the at least one processor and the determined displacement vector corresponding to the moving particle, a plot of a motion particle trajectory of the moving particle.

16. The motion estimator apparatus of claim 4 , wherein the at least one processor is further operative to:

determine displacement by motion for the input image sequence based on the solved symmetric displaced frame difference equation.

17. The non-transitory computer readable medium of claim 7 , further comprising computer-executable instructions for:

determining, using the at least one processor, displacement by motion for the input image sequence based on the solved symmetric displaced frame difference equation.