LOCALIZED AND COMPUTATIONALLY EFFICIENT APPROACH TO SHIFT-VARIANT IMAGE DEBLURRING ICIP, Oct. 2008, San Diego Murali SubbaRao, Youn-sik Kang, Satyaki Dutta*, Xue Tu {murali , yskang, tuxue}@ece.sunysb.edu, *[email protected]Dept. of Electrical and Computer Engineering *Dept. of Mathematics State University of New York at Stony Brook, NY 11794
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LOCALIZED AND COMPUTATIONALLY EFFICIENT APPROACH TO
SHIFT-VARIANT IMAGE DEBLURRING
ICIP, Oct. 2008, San Diego
Murali SubbaRao, Youn-sik Kang, Satyaki Dutta*, Xue Tu
Problems: – k( ) is in global form.– inverting the equation to obtain f( ) is difficult.– Discrete form of the above equation is often used to solve for f( )
in small (e.g. 32x32) image blocks using a Singular Value Decomposition (SVD) technique (e.g. spectral filtering).
– Computational cost is exorbitant, and accuracy may be limited.
( ) ( ) ( )b d
a cg x y k x y u v f u v d u d v, = , , , , .∫ ∫
Rao Transform (RT) Theory and Algorithm
Define a completely localized kernel
Reformulate the problem in an equivalent form that definesRao Transform (RT):
In this form, it becomes possible to invert this integral transform locally under the assumption of analyticity of images.
f( ) is expanded in a Taylor-series at f(x,y), and order or integration and summation are interchanged.
Various order derivatives of g(x,y) are consedered.
( , , , ) ( , , )h x y u v k x u y v x y= + , +
( ) ( ) ( )x a y c
x b y dg x y h x u y v u v f x u y v du dv
− −
− −, = − , − , , − , − .∫ ∫
RT Theory and Algorithm
Blurred image can be expressed as the weighted sum of the derivatives of the focused images where the weights are determined by the given shift-variant PSF (SV-PSF):
By considering the derivatives of g( ), the following linear system of equations can be obtained:
A regularization technique (e.g. SVD based spectral filtering) can be used to obtain a better estimate of inverse RT while inverting . This yields a new localized regularization technique.
, , ,x y x y x y=g R f
, , ,x y x y x y′=f R g 1, ,( )x y x y
−′ =R R
( )
0 0( )
N nn i i
n in i
f x y gS − ,,
= =
, = ′∑∑
,x yR
RT/IRT Example: Symmetric SV-PSFBlurred image:
Closed-form solution to Focused Image:
We are not aware of any previous work that presents such a localized closed-form solution, or one that has similar
Analysis of RT/IRTIf higher order derivatives of g( ) used in the computation of IRT
are erroneous (due to noise, usually so), then, solution for f ( )will have errors, even if a localized regularization technique is used.
The region of significant support of the blurring kernel often satisfies the condition (e.g. blur circle radius or sigma)
The error will be low if, within the region of significant support of the blurring kernel, Taylor-series expansion of f( ) in RT provides a good approximation to the actual product under the integral:
Further research has provided some improved solutions to theseproblems. They will be presented in a future publication.
( ) ( )h x u y v u v f x u y v− , − , , − , −
( ) 0 for | | or | |h x u y v u v u R v R− , − , , ≈ > >
Analysis of Computational Complexity
For a conventional SVD method based on spectral filtering, a 32x32 image requires inverting a matrix, i.e. O(32^6)=O(2^30).
In the RT approach, computations are dominated by the inversionof NxN size matrix , once at each of the 32^2 pixels. N is the order up to which the derivatives of g( ) are used.Setting N=4, the computational complexity becomes O(16^3 X 32^2)=O(2^22).
In this example, approximate computational advantage is a factorof 2^8=256. In our MATLAB implementation, without code optimization, we have observed a speed up of 2 to 4.