Methods and apparatus for computing the input and output signals of a linear shift-variant system

• US 20060101106A1
• Filed: 09/26/2005
• Published: 05/11/2006
• Est. Priority Date: 11/08/2004
• Status: Active Grant

• Current Assignee: Muralidhara Subbarao
• Original Assignee: Muralidhara Subbarao
• Inventors: Subbarao, Muralidhara
• Granted Patent: US 7,558,709 B2
• Time in Patent Office: Days
• US Class Current: 708/400
• CPC Class Codes: G05B 17/02 electric
• CPC Class Codes: G06T 5/002 Denoising; Smoothing noise processing or correction adapted to be used in an image pickup device containing and electronic image sensor H04N23/81, H04N25/60 - H04N25/67
• CPC Class Codes: G06T 5/003 Deblurring; Sharpening vibration or motion blur correction for cameras comprising an electronic image sensor H04N23/682
• CPC Class Codes: G06T 5/20 using local operators
• CPC Class Codes: G06T 5/70 Denoising; Smoothing
• CPC Class Codes: G06T 5/73 Deblurring; Sharpening

Abstract

This invention is based on a new signal processing transform named Rao Transform (RT) which was invented recently by the author of the present invention. Forward RT provides a computationally efficient method and an associated apparatus for computing the output signal of a Linear Shift-Variant System (LSVS) from the input signal and a set of moment parameters of the linear Shift-Variant Point Spread Function (SV-PSF) that characterizes the LSVS. Inverse RT provides a computationally efficient method and an associated apparatus for computing the input signal or restored signal of an LSVS from the output signal and a set of moment parameters of the linear SV-PSF that characterizes the LSVS. This invention is useful in many applications including the restoration of defocus blurred and motion blurred images recorded by a camera with a linear SV-PSF. The apparatus include means for computing forward and inverse RT coefficients.

Claims

1. A method of computing the output signal g(x)of a Linear Shift-Variant System (LSVS) where x is an m-dimensional variable, said output corresponding to an input signal f(x), said LSVS characterized by a Shift-Variant Point Spread Function (SV-PSF) h(x,α

), said method comprising

a. Computing a set of Rao Transform (RT) coefficients S_n of said input signal f(x) with respect to said SV-PSF h(x,α

) wherein said Rao Transform (RT) is defined by an expression that is a definite integral with respect to α

of the product of h(x−

α

,α

) and f(x−

α

), said coefficients S_n obtained by;

1. substituting a truncated Taylor series expansion of h(x−

α

,α

) around the point (x,α

) in said expression that defines RT,

2. substituting a truncated Taylor series expansion of f(x−

α

) around the point x in said expression that defines RT,

3. simplifying said expression after the substitutions above as a sum of a set of product terms wherein each product term is the result of multiplying a derivative of f(x) with respect to x at x and a coefficient that determines S_n and expressed in terms of a set of moments with respect to a of the zero-th and higher order derivatives with respect to x of said SV-PSF h;

b. computing a set of derivatives f⁽ⁿ⁾ of f(x) with respect to x at x of said input signal f(x);

c. computing a set of product terms obtained by multiplying each RT coefficient S_n by its corresponding input signal derivative f⁽ⁿ⁾,

d. summing or adding all members of said set of product terms to obtain the output signal g(x) of said LSVS.

2. The method of claim 1 wherein said input signal is first passed through a noise reduction filter before further processing according to steps (a) through (d).

3. The method of claim 1 wherein said SV-PSF is a Gaussian function.

4. The method of claim 1 wherein said input and output signals are one-dimensional wherein x and a each represent a single scalar variable.

5. The method of claim 1 wherein said input and output signals are two-dimensional signals wherein x and a each represent a vector with two components.

6. The method of claim 5 wherein said two-dimensional input and output signals are images.

7. The method of claim 1 wherein said LSVS is an imaging optical system.

8. A method of computing the input or restored signal f(x) of a Linear Shift-Variant System (LSVS) where x is an m-dimensional variable, said input corresponding to an output or degraded signal g(x), said LSVS characterized by a Shift-Variant Point Spread Function (SV-PSF) h(x,α

), said method comprising

a. Computing a set of Inverse Rao Transform (IRT) coefficients S′

_n corresponding to said output signal g(x) and said SV-PSF h(x,α

) wherein each IRT coefficient S′

_n is determined by

i. considering an expression for said output signal g(x) as a sum of a set of product terms wherein each product term is the result of multiplying a derivative f⁽ⁿ⁾ of said input signal f(x) and a Rao Transform coefficient S_n that corresponds to said derivative,

ii. taking different derivatives of said expression with respect to x and obtaining one equation corresponding to each derivative to obtain a set of equations in which the different derivatives f⁽ⁿ⁾ of said input signal f(x) are the only unknowns, and

iii. solving said set of equations and expressing the solution for the zero-th order derivative of said input signal as a sum of a set of product terms wherein each product term is the result of multiplying a derivative g⁽ⁿ⁾ of said output signal g(x) and a coefficient that is defined to be the Inverse Rao Transform coefficient S′

_n,

b. Computing a set of derivatives with respect to x at x of said output signal g(x),

c. multiplying each IRT coefficient S′

_n by its corresponding output signal derivative g⁽ⁿ⁾ to obtain a set of product terms,

d. summing or adding all members of said set of product terms to obtain the input or restored signal f(x) of said LSVS.

9. The method of claim 8 wherein said output signal is first passed through a noise reduction filter before further processing according to steps (a) through (d).

10. The method of claim 8 wherein said input and output signals are one-dimensional signals wherein x and reach represent a single scalar variable.

11. The method of claim 8 wherein said SV-PSF is a Gaussian function.

12. The method of claim 8 wherein said input and output signals are two-dimensional signals wherein x and a each represent a vector with two components.

13. The method of claim 12 wherein said two-dimensional input and output signals are images.

14. The method of claim 8 wherein said LSVS is an imaging optical system.

15. Apparatus for computing the input or restored signal f(x) of a Linear Shift-Variant System (LSVS) corresponding to an output signal g(x), said LSVS characterized by a Shift-Variant Point Spread Function (SV-PSF) h(x,α

), said apparatus comprising;

i. a differentiation means for computing derivatives of said output signal g(x)to obtain a set of derivative signals,

ii. an Inverse Rao Transform (IRT) coefficient computation means for computing said IRT coefficients S′

_n of said output signal g(x) with respect to said SV-PSF h,

iii. a multiplication means for multiplying each member of said set of derivative signals with its corresponding IRT coefficient S′

_n to obtain a set of product terms, and

iv. a summation means for summing said set of product terms to obtain said input or restored signal.

16. The apparatus of claim 15 wherein computing means are electronic digital devices.

17. The apparatus of claim 15 which further includes a noise reduction means.

18. Apparatus for computing the output signal g(x) of a Linear Shift-Variant System (LSVS) corresponding to an input signal f(x), said LSVS characterized by a Shift-Variant Point Spread Function (SV-PSF) h(x,α

), said apparatus comprising;

i. a differentiation means for computing derivatives of said input signal f(x)to obtain a set of derivative signals f⁽ⁿ⁾,

ii. a Rao Transform (RT) coefficient computation means for computing the RT coefficients S_n of said input signal f(x) with respect to said SV-PSF h,

iii. a multiplication means for multiplying each member of said set of derivative signals f⁽ⁿ⁾ with its corresponding Rao Transform coefficient S_n to obtain a set of product terms, and

iv. a summation means for summing said set of product terms to obtain said output signal.

19. The apparatus of claim 18 wherein computing means are electronic digital devices.

20. The apparatus of claim 18 which further includes a noise reduction means.