As noted in the last post about Frontiers in Optics (FiO), one of the interesting things that will happen at the conference is the special symposium on phase-space optics. I have been intrigued by these ideas since sometime now. In upcoming posts, I will recount some basics of the phase-space optics. I hope those who are getting started in this area will find these posts informative or at least interesting, and those (particularly FiO attendees) who already have some insights will share them via comments.
I have found books by Leon Cohen [1] and a compilation of selected papers as part of SPIE milestone series [2] useful. Cohen has contributed greatly to the basic ideas of phase-space representations and showed that all phase-space distributions are special cases of a particular distribution – which has come to be known as the Cohen class. The phase-space representations have been used in optics (and signal processing in general) for analysis and representation of signals and systems in a way that matches our intuition more closely. However, once someone has been used to the Fourier tools for few years , the phase-space representations may not seem that intuitive, as happened in my case:-).
Joint distributions were first invented in quantum mechanics. They were used by Eugene Wigner [2, pp. 30] to represent probability of a particle possessing given position and given momentum. The distribution represented how particle may be `distributed’ as a function of position and momentum. Gabor [2, pp. 120] and Ville [2, pp. 149] introduced joint distributions to signal analysis to represent temporal signals in a way that matches human intuition. Until the works by Gabor and Ville, signal analysis was performed either in time-domain or in frequency-domain. However, when a human analyzes signal, he/she is usually interested in how signal’s frequency changes over time, e.g., how pitch of some one’s voice changes over time. The last sentence of the abstract of Ville’s paper reads “These notions of instantaneous frequency and of the instantaneous spectrum are introduced to furnish a firm theoretical basis for studies of frequency modulation, …, and in a general way,of all problems for which classical harmonic analysis furnishes a description which departs too far from physical reality.”
A good amount of terminology (e.g., phase-space, distributions, marginals, uncertainty principle) in phase-space signal processing has been borrowed from quantum mechanics, where analogous ideas first originated. I do not say that ideas were similar, as the motivations for using these mathematical constructs are quite different in quantum mechanics and signal processing. As Leon Cohen says in his well-written book [1] “The analogy is only formal [, but not physical].”
Keeping the above point in mind allows us to distinguish physical implications of analogous ideas in both fields. Let’s take an example of the uncertainty principle. In quantum mechanics, uncertainty principle indeed represents uncertainty in being able to measure the properties of a particle. But, in deterministic signal analysis, there is no uncertainty involved. Straightforward extrapolation of the uncertainty principle from quantum mechanics to signal analysis may suggest that one cannot measure the instantaneous spectrum with high temporal resolution. Though plausible, this is not the correct interpretation – for whatever the definition of instantaneous spectrum, one can compute it to arbitrary accuracy since the signal is deterministic. The correct interpretation for signal analysis is that a real signal cannot possess both infinitesimal bandwidth and infinitesimal duration simultaneously. The uncertainty principle in signal processing stems from the fact that the quantities of bandwidth and duration are related to each other via Fourier transform relationship. The analogous principle in quantum mechanics stems from the fact that one cannot measure both position and momentum together with arbitrary accuracy.
I have been wondering about following interesting questions and trying to understand some of the answers with simulations. In upcoming posts, I will share those simulations and ideas.
- What is instantaneous frequency? How does Wigner distribution measure it?
- What do the phase-space co-ordinates of the Wigner distribution and Ambiguity function physically represent?
- How is basic optical processing such as propagation and multiplication by transparency represented in phase-space?
- How is partially coherent field represented in phase-space? (This is the point at which phase-space representation became valuable to me- they provide elegant connections between theory for coherent optical processing and partially coherent optical processing.)
- How about representations of overall transfer properties of partially coherent systems?
Wait a minute, why did I suddenly jump to ‘partially coherent’ signals and systems! Most of the common contrast mechanisms (bright field, phase-contrast, differential interference contrast, differential phase contrast) use large illumination apertures which leads to partially coherent illumination.
If the aperture were a point located on the axis, the illumination received by the specimen will be coherent, i.e., there will be a deterministic relationship among the instantaneous field amplitudes across the entire spatial extent of the specimen. A source with large spatial extent can be thought of as consisting of individually incoherent point sources. Such a source leads to illumination of the specimen by a field which is neither completely random nor completely deterministic. Such fields are called partially coherent fields and these imaging systems partially coherent systems. The large illumination aperture leads to higher spatial resolution, improved depth-sectioning and lack of speckle, in comparison to coherent systems such as laser based holography. I am interested in quantitative imaging by partially coherent systems. Quantitative imaging requires that we have an accurate (which preferably leads to physical insights and is computationally efficient) model for image formation in partially coherent systems. One of my presentations at FiO will discuss our attempt at phase-space description of partially coherent systems.
References
- L. Cohen, Time-frequency analysis: theory and applications (Prentice Hall, 1995).
- M. E Testorf, J. Ojeda-Castañeda, and A. W Lohmann, eds., Selected papers on phase-space optics, vol. 181, SPIE Milestone series, 2006.
4 responses so far ↓
S. Barwick // October 7, 2009 at 2:12 am |
“Straightforward extrapolation of the uncertainty principle from quantum mechanics to signal analysis may suggest that one cannot measure the instantaneous spectrum with high temporal resolution. Though plausible, this is not the correct interpretation – for whatever the definition of instantaneous spectrum, one can compute it to arbitrary accuracy since the signal is deterministic.”
The analogy to quantum may be more apt than you indicate. Mind you, I’m not certain, but here is my interpretation. The uncertainty principle implies that certain properties cannot be measured simultaneously to arbitrary accuracy, not they can’t be measured to arbitrary accuracy individually by separate probes. Right?
Wavelets are a type of phase-space distribution, for example. Individual wavelet atoms are generally interpreted as covering a certain panel in the phase-space plane (1-D signals). A wavelet that is localized in space and, thus, provides precise spatial information, will have a broad spectrum and vice versa. Thus, the chosen phase-space probe will in fact be subject to a restriction akin to the uncertainty principle.
Also, all squared phase-space decompositions like a wavelet scalogram can be shown to be a phase-space averaging of the Wigner-Ville with a smoothing kernel. In support of your interpretation the WVD does, however, satisfy phase and space marginal integrals (integrating wrt phase gives the square of the spatial distribution and vice versa), but the WVD is not strictly positive due to interference terms. In fact, it has been shown that no positive quadratic energy distribution can satisfy the marginal integrals. The required smoothing that reduces phase-space resolution seems to bear some resemblance to the uncertainty principle in QM.
Shalin Mehta // October 7, 2009 at 12:28 pm |
I think that as far as signal analysis is concerned, one can measure both time features and frequency features to better accuracy than uncertainty principle. I am not familiar with quantum mechanical methods of measuring position and momentum simultaneously, so find it hard to visualize what aspect limits the resolution of measurement. The limit on signal measurement will be imposed by capabilities of the measurement instrument. Following example comes to mind: Capture a signal at very high sampling-rate using the best A2D converter and then compute a continuous wavelet transform or continuous wigner distribution in the other dimension. However, if we compare the time-width and band-width from this measurement and analysis, we will find that their product cannot be smaller than uncertainty limit. Thus, the uncertainty principle limits the time-bandwidth product of a signal rather than its measurement. One may think of making the above distinction as pedantic but it seems an important conceptual difference to me.
This is a very interesting point, viz., all physically meaningful distributions are related to Wigner distribution with a smoothing kernel. As an example, we find that the phase-space representation of partially coherent imaging systems is a convolution of the specimen’s Wigner distribution with a kernel dependent on the imaging system’s parameters (illumination and imaging pupils). This is the topic of my talk on Monday at FiO. I think of this situation as implying that signal obtained from a physical system has larger or at best the same time-bandwidth product as obtained with Wigner distribution.
S. Barwick // October 8, 2009 at 2:22 am |
I see your point. The standard interpretation of the uncertainty principle seems to be that the inability to measure inverse quantities simultaneously is a property of the system itself, not the measurement device, which is not true for deterministic signals. By contrast, my analogy (not an exact one) pertains to how the measurement is mathematically made with a particular phase-space distribution. The connection is that both ultimately stem from the time-bandwidth product.
As a sidebar, the correct interpretaion of measurement in QM seems to be disputed, though that discussion is beyond my pay grade. I believe Roger Penrose has a new book coming out that challenges some current orthodoxy. See a recent interview in Discover. Also, I’m wondering how the points on deterministic signals apply to fractals. But that’s a different topic.
Thanks for the enlightment and the blog.
Shalin Mehta // October 8, 2009 at 12:20 pm |
Thanks for thought-provoking comments. Without your comments, this issue wouldn’t be so clear. I find Leon Cohen’s book and papers very readable and insightful for such aspects.