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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.<ref>http://newsroom.ucla.edu/portal/ucla/ucla-research-team-invents-new-249693.aspx</ref><ref>http://www.photonics.com/Article.aspx?AID=55602</ref> 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]]<ref>J. L. Hunt, B. G. Nickel, and C. Gigault, "Anamorphic images", American Journal of Physics 68, 232–237 (2000).</ref> and [[Surrealism|surrealist]] artworks.<ref>Editors of Phaidon Press (2001). "The 20th-Century art book." (Reprinted. ed.). London: Phaidon Press. ISBN 0714835420.</ref><ref>http://www.scienceagogo.com/news/20131120231425.shtml</ref>
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.<ref>{{cite web|url=http://newsroom.ucla.edu/portal/ucla/ucla-research-team-invents-new-249693.aspx|title=New data compression method reduces big-data bottleneck; outperforms, enhances JPEG|author=Matthew Chin|work=UCLA Newsroom}}</ref><ref>{{cite web|url=http://www.photonics.com/Article.aspx?AID=55602|title=‘Warping’ Compresses Big Data|date=30 December 2013|publisher=}}</ref> 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]]<ref>J. L. Hunt, B. G. Nickel, and C. Gigault, "Anamorphic images", American Journal of Physics 68, 232–237 (2000).</ref> and [[Surrealism|surrealist]] artworks.<ref>Editors of Phaidon Press (2001). "The 20th-Century art book." (Reprinted. ed.). London: Phaidon Press. ISBN 0714835420.</ref><ref>{{cite web|url=http://www.scienceagogo.com/news/20131120231425.shtml|title=Radical new data compression method delivers significant gains in quality and speed over existing techniques|publisher=}}</ref>


== Operation principle ==
== Operation principle ==


An anamorphic stretch transform (AST)<ref>M. H. Asghari and B. Jalali, "Anamorphic transformation and its application to time-bandwidth compression", Applied Optics, Vol. 52, pp. 6735-6743 (2013). [http://www.opticsinfobase.org/ao/abstract.cfm?uri=ao-52-27-6735]</ref><ref>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. [http://www.opticsinfobase.org/abstract.cfm?uri=FiO-2013-FW6A.2]</ref> 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]]<ref>L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322</ref><ref>B. Boashash, ed., “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003.</ref><ref>S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.</ref> 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.<ref>J. W. Goodman, "Introduction to Fourier Optics", McGraw-Hill Book Co (1968).</ref>
An anamorphic stretch transform (AST)<ref name="auto">M. H. Asghari and B. Jalali, "Anamorphic transformation and its application to time-bandwidth compression", Applied Optics, Vol. 52, pp. 6735-6743 (2013). [http://www.opticsinfobase.org/ao/abstract.cfm?uri=ao-52-27-6735]</ref><ref>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. [http://www.opticsinfobase.org/abstract.cfm?uri=FiO-2013-FW6A.2]</ref> 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]]<ref>L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322</ref><ref>B. Boashash, ed., “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003.</ref><ref>S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.</ref> 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.<ref>J. W. Goodman, "Introduction to Fourier Optics", McGraw-Hill Book Co (1968).</ref>


The mathematical transformation emulates propagation of the signal through a physical medium with specific dispersion and diffraction properties,<ref>T. Jannson, "Real-time Fourier transformation in dispersive optical fibers", Opt. Lett. 8, 232–234 (1983).</ref><ref>D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves", Nature 450, 1054–1057 (2007).</ref><ref>B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias and J. M. Dudley, "Real-time full bandwidth measurement of spectral noise in supercontinuum generation", Scientific Reports 2, Article number: 882 (2012).</ref> but with an engineered nonlinear kernel.
The mathematical transformation emulates propagation of the signal through a physical medium with specific dispersion and diffraction properties,<ref>T. Jannson, "Real-time Fourier transformation in dispersive optical fibers", Opt. Lett. 8, 232–234 (1983).</ref><ref>D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves", Nature 450, 1054–1057 (2007).</ref><ref>B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias and J. M. Dudley, "Real-time full bandwidth measurement of spectral noise in supercontinuum generation", Scientific Reports 2, Article number: 882 (2012).</ref> but with an engineered nonlinear kernel.
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== Sparsity requirement ==
== Sparsity requirement ==
AST applies a tailored group dispersion to different frequency features.<ref>B. Jalali, J. Chan, and M. H. Asghari, “Time–bandwidth engineering,” Optica 1, 23-31 (2014).
AST applies a tailored group dispersion to different frequency features.<ref>B. Jalali, J. Chan, and M. H. Asghari, “Time–bandwidth engineering,” Optica 1, 23-31 (2014).
</ref><ref>M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time–bandwidth compression,” Appl. Opt. 52, 6735 (2013).</ref><ref>M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression”, Appl. Phys. Lett. 104, 111101 (2014).</ref><ref>J. Chan, A. Mahjoubfar, M. H. Asghari, and B. Jalali, “Reconstruction in time-bandwidth compression systems,” Applied Physics Letters journal, Vol. 105, 221105 (2014).</ref> 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.
</ref><ref>M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time–bandwidth compression,” Appl. Opt. 52, 6735 (2013).</ref><ref>M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression”, Appl. Phys. Lett. 104, 111101 (2014).</ref><ref name="auto1">J. Chan, A. Mahjoubfar, M. H. Asghari, and B. Jalali, “Reconstruction in time-bandwidth compression systems,” Applied Physics Letters journal, Vol. 105, 221105 (2014).</ref> 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.


== Lossy and lossless compression ==
== Lossy and lossless compression ==


The reconstruction accuracy and lossy nature of this compression has been analyzed previously.<ref>J. Chan, A. Mahjoubfar, M. H. Asghari, and B. Jalali, “Reconstruction in time-bandwidth compression systems,” Applied Physics Letters journal, Vol. 105, 221105 (2014).</ref> 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 the original paper <ref>M. H. Asghari and B. Jalali, "Anamorphic transformation and its application to time-bandwidth compression", Applied Optics, Vol. 52, pp. 6735-6743 (2013). [http://www.opticsinfobase.org/ao/abstract.cfm?uri=ao-52-27-6735]</ref>, ideal phase recovery was assumed to show the impact of AST alone. In practice, successful implementation requires phase recovery and this is still an open problem.
The reconstruction accuracy and lossy nature of this compression has been analyzed previously.<ref name="auto1"/> 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 the original paper <ref name="auto"/>, ideal phase recovery was assumed to show the impact of AST alone. In practice, successful implementation requires phase recovery and this is still an open problem.


== Relation to Phase Stretch Transform ==
== Relation to Phase Stretch Transform ==

Revision as of 23:32, 18 January 2016

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][5]

Operation principle

An anamorphic stretch transform (AST)[6][7] 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[8][9][10] 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.[11]

The mathematical transformation emulates propagation of the signal through a physical medium with specific dispersion and diffraction properties,[12][13][14] but with an engineered nonlinear kernel.

Sparsity requirement

AST applies a tailored group dispersion to different frequency features.[15][16][17][18] 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.

Lossy and lossless compression

The reconstruction accuracy and lossy nature of this compression has been analyzed previously.[18] 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 the original paper [6], ideal phase recovery was assumed to show the impact of AST alone. In practice, successful implementation requires phase recovery and this is still an open problem.

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

This transform is a physics-based mathematical operation that reduces the image brightness bandwidth without proportional increase in its size, thus providing space-bandwidth product compression. AST can be used as a pre-processing operation than may enhance image compression techniques such as JPEG, JPEG 2000 or WebP.[19]

Time domain signals

The AST 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.

AST has been experimentally proven to enhance ultrafast signal measurements [20][21] and analog to digital conversion[22][23][24] in terms of operation bandwidth and compression of the volume of the generated data. AST may be a promising solution for the big data problem in rare cancer cell detection systems [25] by compressing the amount of data generated without losing the vital information.

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. ^ "Radical new data compression method delivers significant gains in quality and speed over existing techniques".
  6. ^ a b M. H. Asghari and B. Jalali, "Anamorphic transformation and its application to time-bandwidth compression", Applied Optics, Vol. 52, pp. 6735-6743 (2013). [1]
  7. ^ 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. [2]
  8. ^ L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322
  9. ^ B. Boashash, ed., “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003.
  10. ^ S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
  11. ^ J. W. Goodman, "Introduction to Fourier Optics", McGraw-Hill Book Co (1968).
  12. ^ T. Jannson, "Real-time Fourier transformation in dispersive optical fibers", Opt. Lett. 8, 232–234 (1983).
  13. ^ D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves", Nature 450, 1054–1057 (2007).
  14. ^ B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias and J. M. Dudley, "Real-time full bandwidth measurement of spectral noise in supercontinuum generation", Scientific Reports 2, Article number: 882 (2012).
  15. ^ B. Jalali, J. Chan, and M. H. Asghari, “Time–bandwidth engineering,” Optica 1, 23-31 (2014).
  16. ^ M. H. Asghari and B. Jalali, “Anamorphic transformation and its application to time–bandwidth compression,” Appl. Opt. 52, 6735 (2013).
  17. ^ M. H. Asghari and B. Jalali, “Experimental demonstration of optical real-time data compression”, Appl. Phys. Lett. 104, 111101 (2014).
  18. ^ a b J. Chan, A. Mahjoubfar, M. H. Asghari, and B. Jalali, “Reconstruction in time-bandwidth compression systems,” Applied Physics Letters journal, Vol. 105, 221105 (2014).
  19. ^ 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.
  20. ^ F. Li, Y. Park, and J. Azana, “Linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation”, J. Lightwave Technol. 27, 4623–4633 (2009).
  21. ^ C. Wang and J. P. Yao, "Complete Characterization of an Optical Pulse Based on Temporal Interferometry Using an Unbalanced Temporal Pulse Shaping System", J. Lightwave Technol. 29, 789–800 (2011).
  22. ^ G. C. Valley, "Photonic analog-to-digital converters", Optics Express 15, 1955–1982 (2007).
  23. ^ J. Stigwall and S. Galt, "Signal reconstruction by phase retrieval and optical backpropagation in phase-diverse photonic time-stretch systems", Journal of Lightwave Technology 25, 3017–3027 (2007).
  24. ^ W. Ng, T. D. Rockwood, G. A. Sefler, and G. C. Valley, "Demonstration of a large stretch-ratio (M=41) photonic analog-to-digital converter with 8 ENOB for an input signal bandwidth of 10 GHz", IEEE Photonics Technology Letters 24, 1185–1187 (2012).
  25. ^ K. Goda, A. Ayazi, D. R. Gossett, J. Sadasivam, C. K. Lonappan, E. Sollier, A. M. Fard, S. C. Hur, J. Adam, C. Murray, C. Wang, N. Brackbill, D. Di Carlo, and B. Jalali, "High-throughput single-microparticle imaging flow analyzer", Proc. Nat. Acad. Sci., vol. 109, pp. 11630–11635, 2012.
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