Kriti Sen Sharma Graduate Research Assistant

Deblurring &

Applications in Computed Tomography

Outline

1st Half (15min) Deblurring Basics

2nd Half (15min)

Deblurring in CT

Blurring

Imaging Defects

What if we cannot improve imaging process anymore!!!

Solution

= DEBLURRING (Deconvolution)

Invert the imaging defects MATHEMATICALLY

Examples

Real life photography Acquired Image After Deconvolution

Examples

Astronomical Imaging Acquired Image After Deconvolution

Examples

Microscopic Imaging Acquired Image After Deconvolution

Mathematical Model-1

Imaging Defects

x A

b

b = Ax + n xd = A-1 b

Mathematical Model-2

Imaging Defects

!

"(x)

!

"(x) = l(x)# g(x)!

p(x)

!

g(x)

!

g(x) = p(x)" #(x) + n(x), x $ R2

!

l(x) inverse filter of p(x)

Mathematical Model-2

PSF: Point Spread Function

Deblurring Example-1

Noise = 10-10

Deblurring Example-2

Noise = 10-5

ill-posedness of the Inverse problem

Deblurring Example-3

Noise unknown

Solutions-1

Imaging Defects

x A

b

b = Ax + n xd = A-1 b

Recap

Solutions-1

Truncated Singular Value Decomposition

•  A = U Σ VT = [u1 … uN] diag(s1… sN) [v1 … vN]T

•  Truncation

Ak* = [v1 … vN] diag(1/s1… 1/sN) [u1 … uN] T

•  xk = Ak* b

Solutions-1

Using k = 53 i.e. 53 major singular values used

Visible now?

Solutions-2

Imaging Defects

!

"(x)

!

"(x) = l(x)# g(x)!

p(x)

!

g(x)

!

g(x) = p(x)" #(x) + n(x), x $ R2

!

l(x) inverse filter of p(x)

Recap

Solutions-2

Wiener Filter

!

L(u) =P(u*)

P u( )2

+Snu( )

S" u( )

=P(u*)

P u( )2

+1

SNR(u)

!

P(u)P(u*) = P u( )2

L(u) = P(u)"1 =P(u*)

P u( )2

!

Snu( ) : PSD of noise

S" u( ) : PSD of object

!

L u( ) : PSD of inverse filter l(x)

P u( ) : PSD of blurring filter p(x)

End of Deblurring Basics!!

Now to discuss some real applications of

Deblurring in CT

Jing Wang, Ge Wang, Ming Jiang Blind deblurring of spiral CT images

Based on ENR and Wiener filter Journal of X-Ray Science and Technology – 2005 [previous: IEEE Trans. on Medical Imaging 2003]

Blind Deconvolution

1st Problem: Finding P(u) PSD of p(x) = ? 2nd Problem: Finding SNR(u) SNR at different u = ?

!

L(u) =P(u*)

P u( )2

+Snu( )

S" u( )

=P(u*)

P u( )2

+1

SNR(u)

Solution to 1st Problem

Assume p(x) → Gaussian with σ = ?

Deblur at multiple σ

Find σ that gives best deblurring

How to find best σ: Use ENR

Solution to 2nd Problem

Assume SNR(σ) = k

Find k by phantom studies

ENR

•  Edge to Noise Ration

•  in terms of I-divergence (Information Theoretic approach)

•  Noise effect

•  Edge effect

•  ENR = Edge effect / Noise effect

ENR Maximization Principle

maximize ENR(σ,k)

to get optimal σ

Rollano Hijarrubia et. al. Selective Deblurring for Improved Calcification

Visualization and Quantification in Carotid CT Angiography: Validation Using Micro-CT IEEE Transactions on Medical Imaging 2009

Wiener Filter

•  Two problems to be solved: – 1. Point Spread Function (PSF) = ? – 2. Signal to Noise Ratio (SNR) = ?

Solution to 2nd Problem

•  Phantom designed

•  Scanned •  Reconstructed •  Deblurred at various SNR •  Optimum SNR value chosen

Solution to 1st Problem

•  By measuring PSF of a bead image1

•  Resolution of scanner: 0.3 - 0.4mm Bead size: 0.28mm

[1]. Meinel JF, Wang G, Jiang M, et al. Spatial variation of resolution and noise in multi-detector row spiral CT. Acad Radiol. 2003;10:607– 613.

Selective Deblurring

Axial MIPs (Maximum Intensity Projection) of the original, deconvolved, and restored images of the phantom.

MicroCT Reference

Thanks!

Questions?

Summary of J. Wang et.al. Given g(x)

Wiener Filter at various σ to get λ(x,σ)

ENR(σ) calculated

Max ENR(σ): σOPT

λ(x,σOPT)

Wiener filter SNR parameter chosen from

phantom studies

Edge to Noise Ratio

•  I-divergence

•  Noise effect

•  Edge effect

•  ENR

!

I(u,v) = u(x)logu(x)

v(x)" u(x) " v(x)[ ]

x

#x

#

!

N(",k) = I g,G" # $k g,k,"( )( )

!

E(",k) = I #k g,k,"( ),G" $ #k g,k,"( )( )

!

ENR(",k) =E(",k)

N(",k)