With super-resolution compressive holography we try to address several persistent issues in enhanced image and volume reconstruction; these are: Ill-posed 2D-to-3D volume reconstruction, difficult regularization requiring tuning, super-resolution in the far-field, as well as bringing the computational requirements down towards real-time performance. We show examples of recent innovations and outline some of the challenges for the future.

1. SUPER-RESOLUTION
COMPRESSIVE HOLOGRAPHY
Henrik D. Kjeldsen
henrik.daniel.kjeldsen@cern.ch
CERN IDEASQUARE SEMINAR
20th of February 2015

2. SUPER-RESOLUTION
COMPRESSIVE HOLOGRAPHY
Henrik D. Kjeldsen
henrik.daniel.kjeldsen@cern.ch
CERN IDEASQUARE SEMINAR
20th of February 2015
Let’s Enhance, Duncan Robson, Museum of the Moving Image

3. Participatory Experiment
Who in the audience can name all the clips?
Assumptions:
1. Only a few of you
2. Random seating
Do I need to ask all of you or can I find out with
fewer ”measurements”?
Compressive sensing’s 1st cousin: Group testing
Highlight: 1st rule of compressive sensing:
Assumptions might fail, be careful!

4. Background
Implementation of an
EM generalization of
acoustic holography
Recent work
in experimental
neuroscience

5. Background
SUPER-RESOLUTION: x3 in each spatial mode

6. Background
New measures:
Energy flow vector field
Energy source density
Energy dissipation
New hypothesis:
Energy flow reveals
neural network
causality patterns
phase-locked
average of
previous clip

7. Issues: 1. Ill-posed 2D-to-3D model, poor tomography
2. Difficult regularization, tuning required
Tikhonov, Kyrlov, ℓ2-regularization are resolution limited
Background
Rosen et.al., Optics Express, Vol. 19, Issue 7, 2011
Kjeldsen et. al., Journal of Neuroscience Methods, (in revision)

8. Background
Issues:
1. Ill-posed 2D-to-3D model
2. Difficult regularization
3. No super-resolution in far-field
Current work
in acoustic
holography

9. Compressive Holography
Compressive holography can address all three issues!
Compressive holography is a contradiction in terms..
Compressive sensing raises the issue of prior knowledge or
assumptions about the signal, specifically about the
sparsity (which is a pretty weak assumption).
In general, finding the sparsest solution (ℓ0-optimization) is
NP-hard.
When the signal and measurement bases are uncorrelated
ℓ0  ℓ1, which is tractable, but still quite bad.

10. Compressive Holography
There is a wealth of ℓ1-optimization algorithms, but they
are all iterative, i.e. relatively slow.
Compressive sensing’s 2nd cousin: Tensor completion
Assumptions:
1. Approximate low (multilinear)-rank instead of sparsity
2. Data can be sensed in each mode separately
In the 2D case, that is:
IEEE Transactions on Signal Processing, Vol. 63, Jan. 2015

11. Compressive Holography
Advantages:
1. Non-iterative reconstruction
2. No assumption on sparse basis
3. Advantage of higher dimensions
4. Tuning free regularization

12. Compressive Holography
Example:
Hardware realizations?
!=
Compressive Sensing on a CMOS Separable-Transform Image Sensor, Proceedings of the IEEE, 2010

13. 3D object
0
0.5
1
Compressive Holography
Compressive holographic tomography:
Optimize sparsity on full 3D reconstruction instead
of individual 2D slices to overcome ill-posed
2D-to-3D model.
Rosen et.al., Optics Express, Vol. 19, Issue 7, 2011
Phase of kernel
-2
0
2
Numerical backpropagation
0.5
1
1.5
2
Compressive reconstruction
0.2
0.4
0.6
0.8
Applied Optics, Vol. 50, 2011
Diffracted field
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2

14. Compressive Holography
Compressive holographic tomography:
Work in progress….
Tool to render holograms of
complex 3D scenes to test
detailed tomographic and
volumetric reconstructions

15. Super-resolution Compressive Holography
Challenges:
1. Combine the above solutions in a single framework
We can address all three issues, 2D-to-3D, regularization and far-field super-
resolution, as well as speed, but so far not at the same time.
2. Design and implement appropriate hardware sensors
?

16. Marcus Kaiser and Newcastle
Dynamic Connectome Lab
Miles Whittington and
York Oscillations Group
Gary Green and
York Neuroimaging Centre
Marco Manca and
CERN Medical Applications
Acknowledgements