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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 time stretch 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 the 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 context-aware manner, such that it results in nonuniform Fourier domain sampling. The transformation is defined as:

$AST\left\{{\tilde {E}}_{in}(\omega )\right\}=\int _{-\infty }^{\infty }{{\tilde {E}}_{in}^{*}(\omega ){\tilde {E}}_{in}(\omega +\omega _{m})e^{j\omega _{m}\left[{\frac {\phi (\omega +\omega _{m})-\phi (\omega )}{\omega _{m}}}\right]}d\omega }$

where ${\textstyle {\tilde {E}}_{in}(\omega )}$ is the input optical spectrum, $\phi (\omega )$ is the spectral phase added by AST ( ${\textstyle e^{j\phi (\omega )}}$ being the AST warp kernel), and ${\textstyle \omega }$ and ${\textstyle \omega _{m}}$ denote the optical and envelope modulation frequencies, respectively. The detailed of the reshaping depends on the sparsity and redundancy of the input signal and can be obtained by a mathematical function, which is called "stretched modulation $(S_{M})$ distribution" or "modulation intensity distribution" (not to be confused with a different function of the same name used in mechanical diagnostics).

$S_{M}\left(\omega _{m},T\right)=\int _{-\infty }^{\infty }{{\tilde {E}}_{in}^{*}(\omega ){\tilde {E}}_{in}(\omega +\omega _{m})e^{j\omega _{m}\left[{\frac {\phi (\omega +\omega _{m})-\phi (\omega )}{\omega _{m}}}\right]}e^{j\omega T}d\omega }$

The 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] One can interpret the added phasor term ${\textstyle e^{j\omega T}}$ to represent the effect of a time-shift on the spectral autocorrelation of the signal. As a result, the $S_{M}$ distribution can be used to show the effects of the AST spectral phase $\phi (\omega )$ on the temporal duration and intensity envelope bandwidth of the output signal, which is useful in visualizing the time-bandwidth product of the signal.^[11]

Sparsity requirement

AST applies a tailored group dispersion to different spectral 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 (ADC), similar to the methodology used in time-stretch ADC technology. This property has been called "self-adaptive stretching". 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.^[15] Once the parameters are designed, they do not need to respond to the instantaneous value of the signal. The resulting non-uniform sampling, where information rich portions of the signal are sampled at a 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.^[14]

Limitations and challenges

The reconstruction accuracy and lossy nature of this compression method have 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. Because of this, only modest compression can be achieved in practice.

Alternatively, the reconstruction process can be greatly simplified if the information desired is encoded in the spectral envelope of the input signal instead of the temporal envelope. In such a scenario, the true output can be reconstructed simply by directly de-warping the measured output given the designed warp kernel. This has been achieved experimentally for optical image compression.^[16]

Digital implementation

In the digital implementation of AST (DAST) that is 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. The previous equation for AST can be rewritten in discrete form for DAST as: ${\breve {B}}[n,m]=\left|\sum _{k_{1},k_{2}=-\infty }^{\infty }K\left[n-k_{1},m-k_{2}\right]\cdot B\left[k_{1},k_{2}\right]\right|^{N}$ ,

where ${\textstyle K[n,m]=e^{j\phi [n,m]}}$ is the digital version of the warp kernel. 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 naïve uniform downsampling. This property, known as "feature-selective stretching", 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,^[17] 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 will compromise the quality of phase recovery and data compression. Because of this challenge, practical implementation of anamorphic data compression is not achieved yet. With respect to the challenge of finding the right kernel, an algorithm has recently been reported.^[15]

Similar to the simplified reconstruction approach above mentioned above, a digital implementation for image compression which uses direct warping has also been recently reported.^[18] In this alternative data compression method, information-rich portions of the data are dilated in a process that emulates the effect of group velocity dispersion on temporal signals. With this coding operation, the data can be downsampled at a lower rate than without it, even when considering the overhead in transmitting the warping information. In contrast to previous implementation of the warped stretch compression, here the decoding can be performed without the need of phase recovery.

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 proportionally increasing the size of the signal, thus providing space-bandwidth product compression. Its digital implementation emulates the physical effect by a non-uniform allocation of pixel density. This operation may be used as a pre-processing operation that may enhance conventional image compression techniques.^[19]

Time domain signals

This lossy transformation may make it possible to capture and digitize signals that are faster than the speed of the sensor and the digitizer, and 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. The reconstruction accuracy is subject to the signal to noise ratio and will never be ideal.

References

^ Matthew Chin. "New data compression method reduces big-data bottleneck; outperforms, enhances JPEG". UCLA Newsroom.
^ "'Warping' Compresses Big Data". 30 December 2013.
^ J. L. Hunt, B. G. Nickel, and C. Gigault, "Anamorphic images", American Journal of Physics 68, 232–237 (2000).
^ Editors of Phaidon Press (2001). "The 20th-Century art book." (Reprinted. ed.). London: Phaidon Press. ISBN 0714835420.
^ ^a ^b Asghari, Mohammad H.; Jalali, Bahram (2013-09-16). "Anamorphic transformation and its application to time–bandwidth compression". Applied Optics. 52 (27). The Optical Society: 6735–6743. arXiv:1307.0137. Bibcode:2013ApOpt..52.6735A. doi:10.1364/ao.52.006735. ISSN 1559-128X. PMID 24085172. S2CID 15289361.
^ 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. [1]
^ L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322
^ B. Boashash, ed., "Time-Frequency Signal Analysis and Processing – A Comprehensive Reference", Elsevier Science, Oxford, 2003.
^ S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
^ 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).
^ ^a ^b Jalali, Bahram; Chan, Jacky; Asghari, Mohammad H. (2014-07-22). "Time–bandwidth engineering". Optica. 1 (1). The Optical Society: 23-31. Bibcode:2014Optic...1...23J. doi:10.1364/optica.1.000023. ISSN 2334-2536.
^ Asghari, Mohammad H.; Jalali, Bahram (2013-09-16). "Anamorphic transformation and its application to time–bandwidth compression". Applied Optics. 52 (27). The Optical Society: 6735–6743. arXiv:1307.0137. Bibcode:2013ApOpt..52.6735A. doi:10.1364/ao.52.006735. ISSN 1559-128X. PMID 24085172. S2CID 15289361.
^ Asghari, Mohammad H.; Jalali, Bahram (2014-03-17). "Experimental demonstration of optical real-time data compression". Applied Physics Letters. 104 (11). AIP Publishing: 111101. Bibcode:2014ApPhL.104k1101A. doi:10.1063/1.4868539. ISSN 0003-6951.
^ ^a ^b ^c ^d Chan, J.; Mahjoubfar, A.; Asghari, M.; Jalali, B. (2014). "Reconstruction in time-bandwidth compression systems". Applied Physics Letters. 105 (22). AIP Publishing: 221105. arXiv:1409.0609. Bibcode:2014ApPhL.105v1105C. doi:10.1063/1.4902986. ISSN 0003-6951. S2CID 118955013.
^ ^a ^b Mahjoubfar, Ata; Chen, Claire Lifan; Jalali, Bahram (2015-11-25). "Design of Warped Stretch Transform". Scientific Reports. 5 (1). Springer Science and Business Media LLC: 17148. Bibcode:2015NatSR...517148M. doi:10.1038/srep17148. ISSN 2045-2322. PMC 4658532. PMID 26602458.
^ Chen, Claire Lifan; Mahjoubfar, Ata; Jalali, Bahram (2015-04-23). "Optical Data Compression in Time Stretch Imaging". PLOS ONE. 10 (4): e0125106. Bibcode:2015PLoSO..1025106C. doi:10.1371/journal.pone.0125106. ISSN 1932-6203. PMC 4408077. PMID 25906244.
^ Asghari, M. H.; Jalali, B. (2014). "Discrete Anamorphic Transform for Image Compression". IEEE Signal Processing Letters. 21 (7). Institute of Electrical and Electronics Engineers (IEEE): 829–833. Bibcode:2014ISPL...21..829A. doi:10.1109/lsp.2014.2319586. ISSN 1070-9908. S2CID 17721821.
^ Chan, Jacky C. K.; Mahjoubfar, Ata; Chen, Claire L.; Jalali, Bahram (2016-07-01). "Context-Aware Image Compression". PLOS ONE. 11 (7): e0158201. Bibcode:2016PLoSO..1158201C. doi:10.1371/journal.pone.0158201. ISSN 1932-6203. PMC 4930214. PMID 27367904.
^ 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.

[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.

[auto-5] Asghari, Mohammad H.; Jalali, Bahram (2013-09-16). "Anamorphic transformation and its application to time–bandwidth compression". Applied Optics. 52 (27). The Optical Society: 6735–6743. arXiv:1307.0137. Bibcode:2013ApOpt..52.6735A. doi:10.1364/ao.52.006735. ISSN 1559-128X. PMID 24085172. S2CID 15289361.

[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. [1]

[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).

[:0-11] Jalali, Bahram; Chan, Jacky; Asghari, Mohammad H. (2014-07-22). "Time–bandwidth engineering". Optica. 1 (1). The Optical Society: 23-31. Bibcode:2014Optic...1...23J. doi:10.1364/optica.1.000023. ISSN 2334-2536.

[12] Asghari, Mohammad H.; Jalali, Bahram (2013-09-16). "Anamorphic transformation and its application to time–bandwidth compression". Applied Optics. 52 (27). The Optical Society: 6735–6743. arXiv:1307.0137. Bibcode:2013ApOpt..52.6735A. doi:10.1364/ao.52.006735. ISSN 1559-128X. PMID 24085172. S2CID 15289361.

[13] Asghari, Mohammad H.; Jalali, Bahram (2014-03-17). "Experimental demonstration of optical real-time data compression". Applied Physics Letters. 104 (11). AIP Publishing: 111101. Bibcode:2014ApPhL.104k1101A. doi:10.1063/1.4868539. ISSN 0003-6951.

[auto1-14] Chan, J.; Mahjoubfar, A.; Asghari, M.; Jalali, B. (2014). "Reconstruction in time-bandwidth compression systems". Applied Physics Letters. 105 (22). AIP Publishing: 221105. arXiv:1409.0609. Bibcode:2014ApPhL.105v1105C. doi:10.1063/1.4902986. ISSN 0003-6951. S2CID 118955013.

[:1-15] Mahjoubfar, Ata; Chen, Claire Lifan; Jalali, Bahram (2015-11-25). "Design of Warped Stretch Transform". Scientific Reports. 5 (1). Springer Science and Business Media LLC: 17148. Bibcode:2015NatSR...517148M. doi:10.1038/srep17148. ISSN 2045-2322. PMC 4658532. PMID 26602458.

[16] Chen, Claire Lifan; Mahjoubfar, Ata; Jalali, Bahram (2015-04-23). "Optical Data Compression in Time Stretch Imaging". PLOS ONE. 10 (4): e0125106. Bibcode:2015PLoSO..1025106C. doi:10.1371/journal.pone.0125106. ISSN 1932-6203. PMC 4408077. PMID 25906244.

[17] Asghari, M. H.; Jalali, B. (2014). "Discrete Anamorphic Transform for Image Compression". IEEE Signal Processing Letters. 21 (7). Institute of Electrical and Electronics Engineers (IEEE): 829–833. Bibcode:2014ISPL...21..829A. doi:10.1109/lsp.2014.2319586. ISSN 1070-9908. S2CID 17721821.

[18] Chan, Jacky C. K.; Mahjoubfar, Ata; Chen, Claire L.; Jalali, Bahram (2016-07-01). "Context-Aware Image Compression". PLOS ONE. 11 (7): e0158201. Bibcode:2016PLoSO..1158201C. doi:10.1371/journal.pone.0158201. ISSN 1932-6203. PMC 4930214. PMID 27367904.

[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.

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-An '''anamorphic stretch transform '''('''AST''') 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 [[list of transforms|transform]] that emerged from [[time stretch 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 [[data analysis|analytics]]. The reshaping consists of warped stretching in the 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>
-== Operation principle ==
+==Operation principle==
+An anamorphic stretch transform (AST)<ref name="auto">{{cite journal | last1=Asghari | first1=Mohammad H. | last2=Jalali | first2=Bahram | title=Anamorphic transformation and its application to time–bandwidth compression | journal=Applied Optics | publisher=The Optical Society | volume=52 | issue=27 | date=2013-09-16 | pages=6735–6743 | issn=1559-128X | doi=10.1364/ao.52.006735 | pmid=24085172 | arxiv=1307.0137 | bibcode=2013ApOpt..52.6735A | s2cid=15289361 }}</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 context-aware manner, such that it results in nonuniform Fourier domain sampling. The transformation is defined as:
+<math>AST \left \{\tilde{E}_{in}(\omega) \right \} = \int_{-\infty}^\infty
-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 after [[downsampling]], the volume of data is reduced without loss of pertinent information. The recipe for reshaping is prescribed 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 data volume are modified upon dispersion in the time domain, or upon diffraction in the spatial domain.<ref>J. W. Goodman, "Introduction to Fourier Optics", McGraw-Hill Book Co (1968).</ref> It also provides a blueprint for compressing the data.
+{\tilde{E}_{in}^*(\omega)\tilde{E}_{in}(\omega+\omega_m)
+e^{j\omega_m\left [\frac{\phi(\omega+\omega_m)-\phi(\omega)}{\omega_m} \right ]} d\omega}
+</math>
+where <math display="inline">\tilde{E}_{in}(\omega)</math> is the input optical spectrum, <math>\phi(\omega)</math> is the spectral phase added by AST (<math display="inline">e^{j\phi(\omega)}</math> being the AST warp kernel), and <math display="inline">\omega</math> and <math display="inline">\omega_m</math> denote the optical and envelope modulation frequencies, respectively. The detailed of the reshaping depends on the sparsity and redundancy of the input signal and can be obtained by a mathematical function, which is called "stretched modulation <math>(S_M)</math> distribution" or "modulation intensity distribution" (not to be confused with a different function of the same name used in mechanical diagnostics).
-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.
+<math>S_M \left (\omega_m ,T\right ) = \int_{-\infty}^\infty
-== Sparsity requirement ==
+{\tilde{E}_{in}^*(\omega)\tilde{E}_{in}(\omega+\omega_m)
-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).
+e^{j\omega_m\left [\frac{\phi(\omega+\omega_m)-\phi(\omega)}{\omega_m} \right ]}e^{j\omega T} d\omega}</math>
-</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.
+The 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.<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><ref>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).</ref> One can interpret the added phasor term <math display="inline">e^{j\omega T}</math> to represent the effect of a time-shift on the spectral autocorrelation of the signal. As a result, the <math>S_M</math> distribution can be used to show the effects of the AST spectral phase <math>\phi(\omega)</math> on the temporal duration and intensity envelope bandwidth of the output signal, which is useful in visualizing the time-bandwidth product of the signal.<ref name=":0"/>
-== Applications ==
+==Sparsity requirement==
-=== Image compression ===
+AST applies a tailored group dispersion to different spectral features.<ref name=":0">{{cite journal | last1=Jalali | first1=Bahram | last2=Chan | first2=Jacky | last3=Asghari | first3=Mohammad H. | title=Time–bandwidth engineering | journal=Optica | publisher=The Optical Society | volume=1 | issue=1 | date=2014-07-22 | issn=2334-2536 | doi=10.1364/optica.1.000023 | page=23-31| bibcode=2014Optic...1...23J |doi-access=free}}</ref><ref>{{cite journal | last1=Asghari | first1=Mohammad H. | last2=Jalali | first2=Bahram | title=Anamorphic transformation and its application to time–bandwidth compression | journal=Applied Optics | publisher=The Optical Society | volume=52 | issue=27 | date=2013-09-16 | pages=6735–6743 | issn=1559-128X | doi=10.1364/ao.52.006735 | pmid=24085172 |arxiv=1307.0137| bibcode=2013ApOpt..52.6735A | s2cid=15289361 }}</ref><ref>{{cite journal | last1=Asghari | first1=Mohammad H. | last2=Jalali | first2=Bahram | title=Experimental demonstration of optical real-time data compression | journal=Applied Physics Letters | publisher=AIP Publishing | volume=104 | issue=11 | date=2014-03-17 | issn=0003-6951 | doi=10.1063/1.4868539 | page=111101| bibcode=2014ApPhL.104k1101A }}</ref><ref name="auto1">{{cite journal | last1=Chan | first1=J. | last2=Mahjoubfar | first2=A. | last3=Asghari | first3=M. | last4=Jalali | first4=B. | title=Reconstruction in time-bandwidth compression systems | journal=Applied Physics Letters | publisher=AIP Publishing | volume=105 | issue=22 | year=2014 | issn=0003-6951 | doi=10.1063/1.4902986 | page=221105|arxiv=1409.0609| bibcode=2014ApPhL.105v1105C | s2cid=118955013 }}</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]] (ADC), similar to the methodology used in [[time stretch analog-to-digital converter|time-stretch ADC]] technology. This property has been called "self-adaptive stretching". 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.<ref name=":1"/> Once the parameters are designed, they do not need to respond to the instantaneous value of the signal. The resulting non-uniform sampling, where information rich portions of the signal are sampled at a 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 name="auto1"/>
+==Limitations and challenges==
-The world today is awash in digital information captured by sensors or generated in computer simulations. Coping with the fire hose of digital information in this era requires to develop more efficient methods to compress the volume of data without dropping the vital information. Lossless image compression methods based on reversible algorithms do not provide large compression factors. Lossy image compression methods such as [[JPEG]],<ref>W. B. Pennebaker and J. L. Mitchell, "JPEG still image data compression standard", 3rd ed., Springer, 1993.</ref> [[JPEG 2000]]<ref>A. Skodras, C. Christopoulos, and T. Ebrahimi, "The JPEG 2000 still image compression standard", IEEE Signal Process. Mag., vol. 18, pp. 36–58, 2001.</ref> and [[WebP]],<ref>https://developers.google.com/speed/webp/</ref> provide higher compression factors at the price of losing information. AST is a physics-based mathematical operator that reduces the image brightness bandwidth without proportional increase in its size, thus providing space-bandwidth product compression. It does so by self-adaptive stretching of the image without feature detection or prior knowledge of the image. AST can be operated as a standalone image compression technique or combined with a secondary compression method such as standard JPEG, JPEG 2000 or WebP to improve their performance.<ref>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.</ref>
+The reconstruction accuracy and lossy nature of this compression method have 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. Because of this, only modest compression can be achieved in practice.
-=== Time domain signals ===
+Alternatively, the reconstruction process can be greatly simplified if the information desired is encoded in the spectral envelope of the input signal instead of the temporal envelope. In such a scenario, the true output can be reconstructed simply by directly de-warping the measured output given the designed warp kernel. This has been achieved experimentally for optical image compression.<ref>{{cite journal|last1=Chen|first1=Claire Lifan|last2=Mahjoubfar|first2=Ata|last3=Jalali|first3=Bahram|date=2015-04-23|title=Optical Data Compression in Time Stretch Imaging|journal=PLOS ONE|volume=10|issue=4|pages=e0125106|doi=10.1371/journal.pone.0125106|issn=1932-6203|pmc=4408077|pmid=25906244|bibcode=2015PLoSO..1025106C |doi-access=free}}</ref>
-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 more than coarse features. Upon subsequent sampling, this self-adaptive stretch (SAS) causes more digital samples to be allocated to sharp features where they are needed the most, and fewer to coarse features where they would be redundant.
+==Digital implementation==
-AST has been experimentally proven to enhance ultrafast signal measurements<ref>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).</ref><ref>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).</ref> and analog to digital conversion<ref>G. C. Valley, "Photonic analog-to-digital converters", Optics Express 15, 1955–1982 (2007).</ref><ref>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).</ref><ref>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).</ref> in terms of operation bandwidth and compression of the volume of the generated data. AST is a promising solution for the [[big data]] problem in rare cancer cell detection systems<ref>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.</ref> by compressing the amount of data generated without losing the vital information.
+In the digital implementation of AST (DAST) that is 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. The previous equation for AST can be rewritten in discrete form for DAST as:
+<math>\breve{B}[n,m] = \left|\sum_{k_1,k_2=-\infty}^\infty K\left [n-k_1,m-k_2\right ]\cdot B\left [k_1,k_2\right ]\right|^N</math>,
+where <math display="inline">K[n,m]= e^{j\phi[n,m]}</math> is the digital version of the warp kernel. 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 naïve uniform downsampling. This property, known as "feature-selective stretching", 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<ref name="auto"/> and DAST papers,<ref>{{cite journal |first1=M. H. |last1=Asghari | first2=B. |last2=Jalali |title=Discrete Anamorphic Transform for Image Compression | journal=IEEE Signal Processing Letters | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=21 | issue=7 | year=2014 | issn=1070-9908 | doi=10.1109/lsp.2014.2319586 | pages=829–833|bibcode=2014ISPL...21..829A |s2cid=17721821 | doi-access=free }}</ref> 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).<ref name="auto1"/> Finite SNR will compromise the quality of phase recovery and data compression. Because of this challenge, practical implementation of anamorphic data compression is not achieved yet. With respect to the challenge of finding the right kernel, an algorithm has recently been reported.<ref name=":1">{{cite journal | last1=Mahjoubfar | first1=Ata | last2=Chen | first2=Claire Lifan | last3=Jalali | first3=Bahram | title=Design of Warped Stretch Transform | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=5 | issue=1 | date=2015-11-25 | issn=2045-2322 | doi=10.1038/srep17148 | page=17148| pmid=26602458 | pmc=4658532 | bibcode=2015NatSR...517148M |doi-access=free}}</ref>
-== References ==
+Similar to the simplified reconstruction approach above mentioned above, a digital implementation for image compression which uses direct warping has also been recently reported.<ref>{{cite journal|last1=Chan|first1=Jacky C. K.|last2=Mahjoubfar|first2=Ata|last3=Chen|first3=Claire L.|last4=Jalali|first4=Bahram|date=2016-07-01|title=Context-Aware Image Compression|journal=PLOS ONE|volume=11|issue=7|pages=e0158201|doi=10.1371/journal.pone.0158201|issn=1932-6203|pmc=4930214|pmid=27367904|bibcode = 2016PLoSO..1158201C|doi-access=free}}</ref> In this alternative data compression method, information-rich portions of the data are dilated in a process that emulates the effect of group velocity dispersion on temporal signals. With this coding operation, the data can be downsampled at a lower rate than without it, even when considering the overhead in transmitting the warping information. In contrast to previous implementation of the warped stretch compression, here the decoding can be performed without the need of phase recovery.
-{{reflist}}
+==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 (computer vision)|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 proportionally increasing the size of the signal, thus providing space-bandwidth product compression. Its digital implementation emulates the physical effect by a non-uniform allocation of pixel density. This operation may be used as a pre-processing operation that may enhance conventional image compression techniques.<ref>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.</ref>
+===Time domain signals===
+This lossy transformation may make it possible to capture and digitize signals that are faster than the speed of the sensor and the digitizer, and 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. The reconstruction accuracy is subject to the signal to noise ratio and will never be ideal.
+==See also==
+*[[Phase stretch transform]]
+==References==
+{{Reflist}}
 [[Category:Data compression]]