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HOLOGRAPHIC
IMAGE REPRESENTATIONS
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.
WHAT IS HOLOGRAPHY?WHAT IS HOLOGRAPHY? 22
Holography (όλοσ = all, γράφειν = 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
HOLOGRAPHIC SAMPLINGHOLOGRAPHIC SAMPLING 33
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 1:1 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
:I P R nH I n
:Q P n Q
,i j P
Q
P
HOLOGRAPHIC SAMPLING - RECONSTRUCTIONHOLOGRAPHIC SAMPLING - RECONSTRUCTION 44
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
:nH n Q Q
Original image Reordered pixels
Hologram
Reconstruction Interpolation
Portion
HOLOGRAPHIC SAMPLING - EXAMPLEHOLOGRAPHIC SAMPLING - EXAMPLE 55
DATA DATA DATA INTERP
Original image 50% portion of the
hologram is blacked
After interpolating
missing pixels
HOLOGRAPHIC SAMPLING - EXAMPLEHOLOGRAPHIC SAMPLING - EXAMPLE 66
DATA
100% 25% 5%10%50%
HOLOGRAPHIC SAMPLING – PRO ET CONTRAHOLOGRAPHIC SAMPLING – PRO ET CONTRA 77
ADVANTAGES:
Image quality independent on the
portion location
Plausible results even when
reconstructing from 1-5% of the data
Low computational complexity
DISADVANTAGES:
The need to know the exact portion
location
Inefficient predictive compression
Inefficient DCT-based compression
No straightforward treatment of color
images
HOLOGRAPHIC FOURIER REPRESENTATIONSHOLOGRAPHIC FOURIER REPRESENTATIONS 88
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
,1, , jP x yH u v I x y eF
,P x y
,I x y ,H u v
,I x y ,P x y
IFFT
,H u v
je
real imaginary
HOLOGRAPHIC FOURIER REPRESENTATIONSHOLOGRAPHIC FOURIER REPRESENTATIONS 99
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
, abs ,I x y H u v F
,H u v
,, , *
j P x y x y
LPFI x y I x y e h
,I x y
FFT
,H u v
Abs
HOLOGRAPHIC FOURIER – PRO ET CONTRAHOLOGRAPHIC FOURIER – PRO ET CONTRA 1010
ADVANTAGES:
Image quality independent on the
portion location
No need to know the exact portion
location
Low computational complexity
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
APPLICATIONSAPPLICATIONS 1111
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