ICA4DT WP7: Multiresolution restoration and enhancement of series of MR brain images

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PART 3 Image Enhancement and Interpolation
In the Fourier-encoded in-plane MRI
?
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In-Plane versus Through-Plane

• Superresolution along the z-Axis (through-plane),
  
• e.g. Greenspan 2002, Peeters 2004,

Greenspan H., Oz G., Kiryati N., Peled S, MRI
inter-slice reconstruction using super-resolution
Magnetic Resonance Imaging 20 (2002) 437-446
Peeters R., Kornprobst P., et al. The use of
super-resolution techniques to reduce slice
thickness in functional MRI Internat. J. of
Imaging Systems and Technology, Special issue on
High Resolution Image Reconstruction
14(3)131-136, (2004)

• Superresolution in the xy-Plane (in-plane)
• e.g. controversy Peled-Sheffler 2001,2002,

Scheffler K., Superresolution in MRI? Magnetic
Resonance in Medicine (2002) 48408
Peled S., Yeshurun Y., Superresolution in MRI
application to human white fibre tract
vizualization by diffusion tensor imaging
Magnetic Resonance in Medicine (2001)
4529-35 Peled S., Yeshurun Y., Superresolution
in MRI application to human white fibre tract
vizualization by diffusion tensor imaging
Magnetic Resonance in Medicine (2002) 48409
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Shift and Add - Aliasing and SR

• SR in MR through-plane can be based on
  shift-and-add methodology!

In the down sampled function ?f the high
frequencies Ff(kN) is aliased with the low
frequency Ff(k) Knowing the down-sampled
functions ?f and the shifted sample ?T1f T1/2?f
is sufficient to reconstruct f
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SR in through-plane MRI

• So when we have a stack of images ...

one can reconstruct higher frequencies
through-plane since slices can be acquired in a
overlapping fashion and high-resolution
reconstruction is feasible in the z-direction
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• So what is the problem with Fourier-encoded
  in-plane MRI data?

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The Fourier Problem

• Frequency encoding w(x) xgx
• Phase encoding Df(y) ygyty
• Local Magnetization
• m(x,y,t)r(x,y,t) exp i(xgxtx (t) ygyty (t))
• Total Magnetization
• M(t)?dxdy m(x,y,t) Fr(k(t))
• is essentially the Fourier-transform F of r.

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Remarks 3.1

• If r(x,y,t) is not explicitly depending on time
  the total magnetization M(t) is the Fourier
  transform of r with respect to the sampling
  protocol
• 
  k(t) ? t
• which identifies Fourier modes with sampling
  times.
• If r(x,y,t) is explicitly depending on time a
  single image will be corrupted by motion blur
  (each sampled k-mode corresponds to a slightly
  different object)
• If the gradients g and thus k fluctuate due to
  the imaging-device or due to different magnetic
  susceptibility of tissue the effective k we
  measure is some kdk (r,x,k).

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Shifting an image does not help
?dx f(x-x)exp(ikx) exp(ikx) ?dx
f(x)exp(ikx) FTx f(k) exp(ikx)Ff(k)
Shifting the object in the xy-plane will only
change the Fourier modes by a phase factor and
there is a one-to-one map from the shifted to the
non-shifted k-modes. Rotations similarly allow no
access of higher frequencies, though the map is
not one-to-one any more. There is no new spatial
information to be gained by shifting f in case
the recorder is band-limited.
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• Is this the end of High Resolution image
  reconstruction in Fourier-encoded in-plane MRI?

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• Not Necessarily!
• The question is where does one collect the
  necessary information to imply higher
  frequencies
• and what can we possibly understand under the
  notion Superresolution or High resolution Image
  Reconstruction!

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SR and Image restoration?

• There is no fundamental difference between
  Superresolution and Image-restoration since we
  can always sinc-interpolate any image to a finer
  grid to avoid up and down sampling back and forth
  to proceed with restoration-methodology.
• Various influences corrupt the image. Both motion
  and gradient-fluctuations basically blur the
  images.
• The high frequencies are already uncertain in the
  images as acquired.
• One may demonstrate this by applying the
  Rose-model to Fourier-space.

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Rose Resolution Maps
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• still most of the spatial information is
  contained in the uncertain high frequencies

High-pass ( Filter ) Low-pass
X-space (Rose-images) k-space
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It is clear from the Rose-resolution maps that in
fact the intrinsic resolution Is not identical
with the pixel-resolution of the image but in
fact lower. In the Literature we find methods
that deal with image-reconstruction and
restoration. A.) Linear degrading model of the
Fourier-trafo (e.g.Husse 2003, 2004)
Husse S., Goussard Y., Image Reconstruction in
MRI Regularized Approach by Markov Random Fields
Int. Conf. on Image Proc., Proceedings II-843-6
Vol 3 (2003)
Husse S., Goussard Y., Extended Forms of Geman
Yang Algorithm Application to MRI Reconstruction
Proc. IEEE ICASSP Vol III (May 2004) 513-516
B.) Deblurring and Image-Restoration in the
Irani-Peleg spirit. (e.g. Irani Peleg 1990)
Irani M. Peleg S., Improving Resolution by Image
Registration, Graphical Models and Image
Processing, 53, 3, (1990) 231-239
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Deblurring with Irani-Peleg-Type of
Back-Projection

• Guessing the point-spread function equals
  providing prior knowledge. When the results do
  not match our expectations it is not necessarily
  the concept that is wrong but our assumptions.

Rem We get some enhancement of features and an
increase in contrast. However one has to be aware
of limit cycles.
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• Deblurring each image separately increases the
  Rose-resolution of the images

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• However, this has not provided us with
  information on higher frequencies since
  Irani-Peleg type of algorithms can not recover
  frequencies that are zero from the beginning.

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Remaining Options for SR

• Guessing high frequencies with maxent like
  principles.
• e.g Glasbey 2001

Glasbey C.A., Optimal linear interpolation of
images with known point spread function. SCIA
2001, 161-168.

• Training Neural Networks, Kernels, Lookup-tables,
  etc., on image-class information
• e.g Miravet 2005, Candocia 2000,

F.M. Candocia, J.C. Principie Superresolution of
images with learned multiple reconstruction
kernels. Multimedia Image and Video Processing,
Ling Guan et al., (eds), CRC Press, Chapter 4,
pp. 67-95, 2000
Miravet C., Rodriguez F.B. Accurate and robust
image superresolu-tion by neural processing of
local image representations. Lect. Notes Comput.
Sc. 3696 (2005) 499-506

• Other ways of providing valid prior information I
  for the image-representation problem I ? P(rI)
  , like for instance the same scene captured in a
  different modality, tissue-classification,

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Are there still other options?

• Providing valid prior information is a key-issue,
  yet this can only be achieved by a closer
  understanding of the problem at hand and the
  things we actually see, despite the categories we
  might initially be thinking in.
• A new acquisition based ansatz to SR in MRI
  which goes beyond the scope of this presentation
  is taking shape.
• We hope to present it here soon.