1. 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.

2. WHAT IS HOLOGRAPHY? 2 2 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

3. HOLOGRAPHIC SAMPLING 3 3 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

4. HOLOGRAPHIC SAMPLING - RECONSTRUCTION 4 4 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 imageReordered pixels Hologram ReconstructionInterpolation Portion

5. HOLOGRAPHIC SAMPLING - EXAMPLE 5 5 DATA INTERP Original image50% portion of the hologram is blacked After interpolating missing pixels

6. HOLOGRAPHIC SAMPLING - EXAMPLE 6 6 DATA 100%25%5%10% 50%

7. HOLOGRAPHIC SAMPLING – PRO ET CONTRA 7 7 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

8. HOLOGRAPHIC FOURIER REPRESENTATIONS 8 8 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 real imaginary

9. HOLOGRAPHIC FOURIER REPRESENTATIONS 9 9 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

10. HOLOGRAPHIC FOURIER – PRO ET CONTRA 10 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

11. APPLICATIONS 11 ProgressiveProgressive encoding and transmission of images in a distributed environment Data sharing & protectionData 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 transmissionRobust 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 hidingData 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 multiplexingImage multiplexing: storing and transmitting several images simultaneously as a single image

12. PROGRESSIVE ENCODING & TRANSMISSION Progressive transmission of an image SERVER 1 SERVER 2 SERVER 3 CLIENT Portion 1 Portion 2 Portion 3 Portions arrive in random order

13. PROGRESSIVE ENCODING & TRANSMISSION

14. DATA SHARING & PROTECTION PARTY 1 PARTY 2 PARTY 3FULL-QUALITY IMAGE

15. ROBUST STORAGE & TRANSMISSION IMAGE HOLOGRAM NOISE Median filteringDamaged image

16. DATA HIDING Noise removalHidden image IMAGENOISE Image Random noise Missing pixels interpolation

17. IMAGE MULTIPLEXING MUX SOURCE 1 SOURCE 2 SOURCE 3 SOURCE 4 DEMUX