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