HOLOGRAPHIC

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HOLOGRAPHIC IMAGE REPRESENTATIONS
Alexander Bronstein
Based on A.M. Bruckstein, R. J. Holt and A. N.
Netravali, Holographic representation of
images, IEEE Transactions on Image Processing,
Vol 7(11), pp. 1583-1597, 1998.
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WHAT IS HOLOGRAPHY?
Holography (???s all, ???fe?? write) An
optical method of recording a complete
interference pattern of two laser beams targeted
onto an object Every point of a hologram
contains information about the entire
scene IMPORTANT PROPERTY Even from a small
portion of the hologram one can restore the
entire scene The quality depends on the portion
size but not on the portion location
Hologram interference pattern
Reconstructed scene
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HOLOGRAPHIC SAMPLING
IDEA Reorder the pixels of the image and produce
a vector, every portion of which will contain
pixels from the entire image domain with nearly
equal probability. Given an image
produce a vector
is a 11 hash function,
which maps an integer index into a pair
of pixel coordinates The image of by
is a pseudo-random sequence, distributed
approx. uniformly over
Regular pixel ordering
Holographic sampling
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HOLOGRAPHIC SAMPLING - RECONSTRUCTION
Reconstruction is carried out by taking an
arbitrary portion of the hologram and mapping it
back into the image domain Missing pixels are
filled using interpolation
Original image
Reordered pixels
Reconstruction
Interpolation
Portion
Hologram
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HOLOGRAPHIC SAMPLING - EXAMPLE
DATA
DATA
INTERP
DATA
Original image
50 portion of the hologram is blacked
After interpolating missing pixels
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HOLOGRAPHIC SAMPLING - EXAMPLE
DATA
100
25
5
10
50
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HOLOGRAPHIC SAMPLING PRO ET CONTRA
DISADVANTAGES The need to know the exact
portion location Inefficient predictive
compression Inefficient DCT-based
compression No straightforward treatment of
color images
ADVANTAGES Image quality independent on the
portion location Plausible results even when
reconstructing from 1-5 of the data Low
computational complexity

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HOLOGRAPHIC FOURIER REPRESENTATIONS
IDEA Embed the image as the magnitude of a
complex random-phase image. The hologram is
obtained by the inverse Fourier transform
where is a random i.i.d. phase
with uniform distribution. Random phase
spreads the information about the image
all over

IFFT
imaginary
real
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HOLOGRAPHIC FOURIER REPRESENTATIONS
Reconstruction from a portion of is performed by
taking the magnitude of the Fourier
transform The restored image is where
and depend on the portion location Cut-off
frequency of the LP filter is inverse
proportional to the portion size No need to know
the portion location

FFT
Abs
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HOLOGRAPHIC FOURIER PRO ET CONTRA
DISADVANTAGES Poor reconstruction results even
from 50 of data Inefficient predictive
compression Inefficient DCT-based
compression No straightforward treatment of
color images Complex image doubles the amount of
data Sensitive to quantization

ADVANTAGES Image quality independent on the
portion location No need to know the exact
portion location Low computational complexity

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APPLICATIONS
Progressive encoding and transmission of images
in a distributed environment Data sharing
protection sharing portions of the hologram
between sides, who must agree to collaborate in
order to restore the full-quality image Robust
and failure proof data storage and transmission.
Damage to a contiguous block of pixels in the
hologram has less a destructive effect on the
resulting image Data hiding embed the image
into a pattern of random noise using holographic
sampling. Restoration is possible by whom who
knows the location, at which the image portion
was embedded Image multiplexing storing and
transmitting several images simultaneously as a
single image
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PROGRESSIVE ENCODING TRANSMISSION
Portion 1
Portions arrive in random order
CLIENT
SERVER 1
Portion 2
SERVER 2
Portion 3
Progressive transmission of an image
SERVER 3
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PROGRESSIVE ENCODING TRANSMISSION
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DATA SHARING PROTECTION
PARTY 2
PARTY 3
FULL-QUALITY IMAGE
PARTY 1
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ROBUST STORAGE TRANSMISSION
NOISE
IMAGE
Damaged image
HOLOGRAM
Median filtering
Damaged image
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DATA HIDING
Image
Missing pixels interpolation
Noise removal
Hidden image
IMAGE
NOISE
Random noise
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IMAGE MULTIPLEXING
SOURCE 1
SOURCE 2
SOURCE 3
SOURCE 4
MUX
DEMUX