HOLOGRAPHIC IMAGE REPRESENTATIONS

1. 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. 2 WHAT IS HOLOGRAPHY? 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. 3 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 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. 4 HOLOGRAPHIC SAMPLING - RECONSTRUCTION Reconstruction is carried out by taking an arbitrary portion of the hologramand mapping it back into the image domain Missing pixels are filled using interpolation Original image Reordered pixels Reconstruction Interpolation Portion Hologram

5. 5 HOLOGRAPHIC SAMPLING - EXAMPLE DATA DATA INTERP DATA Original image 50% portion of the hologram is blacked After interpolating missing pixels

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

7. 7 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

8. 8 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

9. 9 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

10. 10 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

11. 11 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 proofdata 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