Anamorphic stretch transform

An anamorphic stretch transform (AST) also referred to as warped stretch transform is a physics-inspired signal transform that emerged from photonic time stretch and dispersive Fourier transform. The transform can be applied to analog temporal signals such as communication signals, or to digital spatial data such as images.[1][2] The transform reshapes the data in such a way that its output has properties conducive for data compression and analytics. The reshaping consists of warped stretching in Fourier domain. The name “Anamorphic” is used because of the metaphoric analogy between the warped stretch operation and warping of images in anamorphosis[3] and surrealist artworks.[4]

Operation principle

An anamorphic stretch transform (AST)[5][6] is a mathematical transformation in which analog or digital data is stretched and warped in a specific fashion such that it results in nonuniform Fourier domain sampling. The detailed of the reshaping depends on the sparsity and redundancy of the input signal and can be obtained by a mathematical function called stretched modulation distribution or modulation intensity distribution (not to be confused with a different function of the same name used in mechanical diagnostics). A stretched modulation distribution is a 3D representation of a type of bilinear time–frequency distribution similar, but not the same, as other time-frequency distributions [7][8][9][10]

Sparsity requirement

AST applies a tailored group dispersion to different frequency features.[11][12][13][14] By matching the group delay dispersion to the spectrum of the particular signal of interest, it performs frequency to time mapping in a tailored fashion. Information rich portions of the spectrum are stretched in time more than sparse regions of the spectrum making them easier to capture with a real-time analog to digital converter. This property has been called self-adaptive stretch. Because the operation is specific to the spectrum of the signal, it does not require knowledge about the instantaneous time domain behavior of the signal. Hence no real-time adaptive control is needed. The parameters of AST are designed using the statistical spectral (not instantaneous) property of signal family of interest in the target application. Once the parameters are designed, they do not need to respond to instantaneous value of the signal. The resulting non-uniform sampling, where information rich portions of the signal are sampled at higher rate than the sparse regions, can be exploited for data compression. As any other data compression method, the maximum compression that can be achieved using AST is signal dependent.

Limitations and Challenges

The reconstruction accuracy and lossy nature of this compression has been analyzed previously.[14] The system reshapes the spectro-temporal structure of the signal such that nearly all the signal energy is within the bandwidth of the real-time digitizer of the acquisition system. Because of the limited bandwidth and the limited resolution of the digitizer, as measured by its effective number of bits (ENOB), the reconstruction will never be ideal, and therefore, this is a lossy compression method.

In digital implementation of AST (DAST) performed in 2D and applied to digital images, an appropriately designed warp kernel stretches the input in a way that reduces the overall spatial bandwidth and hence the sampling requirement requirement. Similar to the case of 1-D temporal waveforms, the warped waveform can then be sampled at a lower rate than what was previously possible with naive uniform downsampling. This property, known as feature selective stretch, can be used for digital image compression. There are two challenges in DAST, (1) image reconstruction, and (2) design the warping kernel. The warped mapping is typically performed in the frequency domain. Reconstruction (inverse mapping) of the spatial image via Fourier transform require knowledge of phase in addition to amplitude of the warped image. In the original AST[5] and DAST papers,[15] ideal phase recovery was assumed to show the useful impact of warp transformation. However,as mentioned above, it has also been shown that phase recovery and signal reconstruction depends on the signal to noise ratio (SNR).[14] Finite SNR can then compromise the quality of phase recovery and this impacts the application of warped stretch to data compression. In practice, successful implementation AST for temporal signals, and DAST for digital images, requires phase recovery and this is still an open area for research. With respect to kernel design, an algorithm for doing this has recently been reported.[16]

Relation to Phase Stretch Transform

The Phase Stretch Transform or PST is a computational approach to signal and image processing. One of its utilities is for feature detection and classification. Both Phase Stretch Transform and AST transform the image by emulating propagation through a diffractive medium with engineered 3D dispersive property (refractive index). The difference between the two mathematical operations is that AST uses the magnitude of the complex amplitude after transformation but Phase Stretch Transform employs the phase of the complex amplitude after transformation. Also, the details of the filter kernel are different in the two cases.

Applications

Image compression

Anamorphic (warped) stretch transform is a physics-based mathematical operation that reduces the signal bandwidth without proportional increase in its size, thus providing space-bandwidth product compression. It's digital implementation can be used as a pre-processing operation than may enhance conventional image compression techniques.[17]

Time domain signals

The technology makes it possible to not only capture and digitize signals that are faster than the speed of the sensor and the digitizer, but also to minimize the volume of the data generated in the process. The transformation causes the signal to be reshaped is such a way that sharp features are stretched (in Fourier domain) more than coarse features. Upon subsequent uniform sampling this causes more digital samples to be allocated to sharp spectral features where they are needed the most, and fewer to sparse portions of the spectrum where they would be redundant.

See also

References

  1. ↑ Matthew Chin. "New data compression method reduces big-data bottleneck; outperforms, enhances JPEG". UCLA Newsroom.
  2. ↑ "‘Warping’ Compresses Big Data". 30 December 2013.
  3. ↑ J. L. Hunt, B. G. Nickel, and C. Gigault, "Anamorphic images", American Journal of Physics 68, 232–237 (2000).
  4. ↑ Editors of Phaidon Press (2001). "The 20th-Century art book." (Reprinted. ed.). London: Phaidon Press. ISBN 0714835420.
  5. 1 2 M. H. Asghari and B. Jalali, "Anamorphic transformation and its application to time-bandwidth compression", Applied Optics, Vol. 52, pp. 6735-6743 (2013).
  6. ↑ M. H. Asghari and B. Jalali, "Demonstration of analog time-bandwidth compression using anamorphic stretch transform", Frontiers in Optics (FIO 2013), Paper: FW6A.2, Orlando, USA.
  7. ↑ L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322
  8. ↑ B. Boashash, ed., “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003.
  9. ↑ S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
  10. ↑ J. W. Goodman, that describes the dependence of intensity or power on the frequency and time duration of the modulation. It provides insight on how information bandwidth and signal duration are modified upon nonlinear dispersion in the time domain, or upon nonlinear diffraction in the spatial domain. "Introduction to Fourier Optics", McGraw-Hill Book Co (1968).
  11. ↑ B. Jalali, J. Chan, and M. H. Asghari, “Time–bandwidth engineering,” Optica 1, 23-31 (2014).
  12. ↑ M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time–bandwidth compression,” Appl. Opt. 52, 6735 (2013).
  13. ↑ M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression”, Appl. Phys. Lett. 104, 111101 (2014).
  14. 1 2 3 J. Chan, A. Mahjoubfar, M. H. Asghari, and B. Jalali, “Reconstruction in time-bandwidth compression systems,” Applied Physics Letters journal, Vol. 105, 221105 (2014).
  15. ↑ M. H. Asghari and B. Jalali, “Discrete Anamorphic transform for image compression,” IEEE Signal Processing Letters, Vol. 21, pp. 829-833 (2014).
  16. ↑ A. Mahjoubfar, C. Lifan Chen, and B. Jalali, "Design of Warped Stretch Transform," Scientific Report 2015; 5: 17148.
  17. ↑ M. H. Asghari and B. Jalali, "Image compression using the feature-selective stretch transform", 13th IEEE International Symposium on Signal Processing and Information Technology (ISSPIT 2013), Athens, Greece.
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