Scintillate, scintillate, globule vivific!

Have a good read of this:
“Scintillate, scintillate, globule vivific!
Fain would I fathom thy nature specific,
Distantly poised in the ether capacious,
Closely resembling a gem carbonaceous. ”

Some of you may already have smiled broadly. If you haven’t yet, see this to find out the meaning of above.

Interplay of fft, ifft, fftshift, and ifftshift in MATLAB

Contents

• The problem – Fourier transform of a real and even function.
• Let us take a concrete example – circular pupil
• First attempt at obtaining the point spread function.
• Second attempt – change the origin before ifft2
• The prescribed use
• Where was the confusion?
• The confusion resolved
• After-thoughts

Hello world,

I routinely simulate forward imaging in optical systems to understand how they function. One of the beautiful phenomena of optics is that a lens is a fourier transformer. It produces 2D fourier transform of field between planes located one focal length away on either side of itself. MATLAB is a ‘matrix laboratory’, so it implements discrete Fourier transform (DFT). The standard way of implementing DFT is FFT (Fast fourier transform) algorithm.

Although I have been using MATLAB since 6 years, I never quite grasped the interplay of fft, ifft, fftshift, and ifftshift functions found in MATLAB. Well, the penny dropped today after a comment from Steve on his blog that fftshift and ifftshift cause circular shifts and nothing more. By the way, Steve’s image processing blog is one of the most useful blogs and these days he is clarifying how Fourier transforms work in MATLAB. He just described that FFT algorithm inherently assumes that the function is periodic in both space and frequency. This occurs because whenever one of the domain (space or spatial-frequency) is sampled, its Fourier counterpart becomes periodic. Multiplication with impulse train (i.e., discretization) in one domain leads to convolution with impulse train in the other domain (i.e., periodicity). In DFT, both space and spatial-frequency are discrete (actually digital), and therefore, both domains are periodic.

The problem – Fourier transform of a real and even function.

Typical functions that occur in optics (e.g., pupil of an imaging system which is simply a circle) are real and even around origin (i.e., optical axis). We expect the Fourier transform or inverse Fourier transform of a real and even function to be real and even itself. However, I had trouble achieving this seemingly basic task. After good amount of experimentation, I had decided that the the idiom spec = fftshift(fft2(ifftshift(signal))) gives real and even spectrum for real and even signal. However, I understood why that is the case just recently.

Wrap-up of FiO/LS 09

The last day was fun. Well, not because I was waiting for the conference to end; but because apart from interesting talks, I had great time meeting Jim Fienup’s group over lunch and guys from different labs before we left San Jose.

I had good fun attending Wonshik Choi’s and Laura’s talks about phase-retrieval. Wonshik’s invited talk (FThB3) was about diffraction tomography (where you indeed measure phase information along many angles in contrast to intensity diffraction tomography I mentioned in previous post). What impressed me was that he understood and explained all assumptions involved in acquisition and reconstruction in simple terms. Laura extended transport of intensity equation (TIE) in two interesting directions. TIE relates axial derivative of intensity of  a beam to transverse distribution of phase as it propagates. By measuring intensity at two slightly defocused planes, one can retrieve specimen phase using TIE. One of her talk (CThA3) showed how one can include  higher-order derivatives of the axial intensity in this formulation. In the other (rather cool) talk (FThR3), she showed how one can use off-the-shelf color camera and exploit chromatic axial shifts  introduced by imaging system to record intensities at two slightly defocused planes. This makes it possible to perform TIE with single snapshot.

Over lunch, I met  Jim Fienup and his group and learned some new things about phase-retrieval and Rochester. Just before we left the conference venue, Laura & Lei (MIT), Naveen & I (Singapore) and Mayukh (Rochester) sat down to chat about peculiarities of our countries and (ahem!) our advisers. Mayukh, who works with Prof. Emil Wolf recounted many interesting stories that Emil has recounted to him. We had stories heard from Colin to share. So the day ended with healthy dose of laugh.

This ends my first trip to the US and I had a great 12-day stay here. I have written this and the previous posts from the airport and I think everyone else is traveling too. Who wants to stay back in San Jose:)? So bon voyage.

Wednesday at FiO – More computational imaging, phase-space anlaysis, and phase-retrieval

I know this post is late, but I hope it is worth the wait (if anyone was waiting ). My energy levels were just not sufficient for me to write anything coherent at the end of the fourth day of conference, but I am feeling better now that the conference has ended. So let’s begin with events of Wednesday.

Wednesday’s program convinced me that parallel universes do exist. I was shuttling between ground floor and first floor (aka banquet floor) trying to pop-in and pop-out of talks at right times. Keeping with theme of last post, I will point out one talk each from above three areas. As a bonus, I also point out how my talk went.

Computational imaging:

The first session of the day that I went to was the joint interdisciplinary session of AO/COSI/SRS. The talk that I remember clearly is Marc Levoy’s on light-field photography and microscopy (JWA3). Marc has visited Singapore earlier and given a similar talk. So he had advised me against attending his talk and spend time gathering something new. But it was good to see how he is planning to apply these methods to decoding of neuronal wiring in zebra fish (or another?) embryos. The idea is to stimulate neurons located at arbitrary 3D locations using light-field illumination architecture. This ’stimulate firing’ of neuron should be followed by fast 3D imaging of activation of neighboring neurons. Yes, the neurons have to be labelled correctly – you need to label neuronal ion channels (rhodopsin) in such a way that when the label is excited the channel opens. Plus, all neurons have to be labelled with calcium indicator dye which indicates when certain neuron becomes active. The enabling technology in labeling is genetic modification of the zebra fish to express these proteins in live animals.

Phase retrieval:

The next noteworthy talk I attended was Greg Gbur’s (SWA1) on intensity diffraction tomography. The idea is rather neat – you measure the phase of the specimen by tomography but without explicitly measuring phase (e.g. by interferometry or otherwise). The idea relies on the fact that phase information of the specimen does affect intensity of diffracted light and hence one ought to be able to retrieve the phase distribution from intensity measurements alone. I should note that similar goal is achieved in a different way by transport of intensity formulation.

In the same session was my talk about registration of gradient information (SWA4). We know that traditional methods of image registration are meant to register measurements of the same function. Therefore, when we want to register gradients of some function along X and Y directions, these methods don’t work. I figured that registration of gradients requires that you reformulate the registration problem and devise a new procedure for computing the registration information. This registration has allowed me to accurately reconstruct phase information from phase-gradients that I measure with differential interference contrast (DIC) and differential phase contrast (DPC). I did better at this talk than the one on Monday and had some useful discussion after it. I was helped by the fact that the speaker after me had withdrawn, giving me more time to delve onto details.

Phase-space analysis:

It was pointed out by Prof. William Rhodes that Wigner distribution computed from discrete samples of a signal may contain erroneous structures (FWW1). This happens because when a temporal signal is sampled, it becomes periodic in frequency with the period equal to the sampling frequency. These periodic components of frequency produce cross-terms. These cross-terms cancel out if one projects the signal along frequency (to obtain signal intensity) but do not if one projects the signal along space (to obtain spectrum density). These errors are exacerbated when one operates on this distribution (e.g. propogate by shearing) and then again tries to compute intensity or spectral density by projection.

FiO/LS Day 3: Computational photography, phase-space analysis and phase-retrieval

Much to the chagrin of FiO attendees, Tuesday morning started with gusty winds and rain. I and my friend had moved to a hotel 1 mile away from the conference venue. Deciding the best way to reach the venue in the down-pour took time and we missed Ramesh Raskar’s talk on computational photography

After finally making to the venue, I attended talks related to computational photography and microscopy, phase-space analysis and phase-retrieval. The trend I am likely to continue for the rest of the two days. There were quite a few I had to miss due to overlap. FiO is a great place to gather ideas from diverse backgrounds. I attended talks from computer scientists designing new computational photography methods, optical microscopists designing computational microscopy methods based on SLM, mathematicians pondering at basics of compressive sensing and phase-space analysis, and astronomers trying to retrieve phase-information about medium between earth and heavenly bodies. Such a mix has to be hard to find elsewhere.

It is interesting to ponder upon how these fields are inter-related: Computational microscopy and photography methods solve problems that are hard to solve with the traditional scheme of ‘acquire first and process later’. Some of these techniques benefit from acquiring light-fields rather than images. Light-field is the distribution of intensity along space and angle and as we know an image is the distribution of intensity only along space. This notion makes it attractive for researchers in computational work to adopt phase-space analysis, which in effect describes optical phenomena in space and angle dimensions. Even though physical acquisition of light-fields may not be involved, there are problems where computation in phase-space *enables* processing tasks that cannot be performed otherwise. After one has acquired non-image but information-rich light-field (or other representation), one is faced with the task of recovering useful information. This is where reconstructions in computational methods and traditional phase-retrieval approaches from optics have parallels.

Let me note just one example from each field that illustrates above connections. These examples were picked because I relatively understood these ideas better than the others.

In the morning session, Fredo Durand (session: CTuB1), a computer scientist from MIT, gave an interesting talk about how one can think of variation of intensities in space and time in terms of light-fields. He took an example of deblurring one dimensional motion, which could be deblurred by imposing parabolic motion on the camera itself.

Immediately following that talk was the talk by Markus Testorf (CTuB2) in which he described how phase-space analysis can be used in design of computational imaging systems. He revived a notion of ‘The instrument function’ which describes what region of phase-space is acquired by an instrument.

In the afternoon was the talk by Monica Ritsch-Marte (FTuU1), in which she described several ways in which a spatial light modulator can be used in microscopic systems to design novel contrasts and to achieve phase-retrieval. Determining patterns that should be put on SLM to achieve certain point spread function of the imaging system and carrying out phase-retrieval requires use of algorithms developed in astronomy (e.g., Gerchberg – Saxton algorithm).

The day 2 – fascinating plenaries, informative sessions and student reception

The second day at FiO turned busy and occupying. So busy that at the end of the day, I had to crash in the bed. This blog is therefore coming little late.

Bob has provided great near-live update of the plenary talks. The plenaries were really fascinating and enthusiasm of plenary speakers really palpable. Prof. Andrea Ghez made one of the best presentation I have ever seen and I hope to emulate her in future. The slides didn’t have more than 5 lines of text and a single graphic, which kept audience focused on what she was saying. I learned for the first time that there is likely a black hole at the center of every galaxy and its region of influence is called Schwarzchild radius. It is the radius below which a given amount of mass will collapse under the influence of gravity. It was fascinating to learn that the earth’s Schwarzchild radius is of the order of sugar cube and of the sun is of the order of a small university campus. However, I didn’t follow the notion that some galaxies have active nuclei that emit enormous jets of gas from their centers. If there is a black-hole at the center, how does that gas escape? Perhaps this gas escapes while whirling around the black hole at the edge of the Schwarzchild region. These ideas were followed by a discussion about the development of adaptive optics and impressive improvements that it provides when imaging stars.

Dr. Janos Kirz started from the origins of X-ray microscopy and brought the audience up to date with the state of the art. In effect, he summarized the key developments of the whole field in half an hour! It was curious to know that the refractive index of  some materials at  X-ray wavelengths is less than 1 and hence one can have total external reflection – a situation that I think does not exist (yet) in optical regime. He described idea of lens-less imaging in very intuitive terms. Basically, a lens can be thought of as returning us an object from the diffraction pattern. But if one records a diffraction pattern, the object can be retrieved by computation – bypassing practical issues of making a lens in X-ray regime.

Afterward, I attended sessions related to biomedical optics at Glen Ellen and gave a talk (about phase-space models for partially coherent systems) in the later half of the afternoon. All talks were interesting and some were really informative. It was great to see how non-linear optical method of CARS is giving molecular information without the need of labels. Several presentations were about design, implementation and application of new microscopic contrast mechanisms. I felt my presentation was well received and I could convey the big picture. But perhaps I put forward too many details. I felt so because some in the audience asked me to send more information rather than asking questions themselves. I hope to do better  at my next talk on Wednesday.

The day ended with a relaxing time at O’ Flaherty’s pub – the venue  of OSA student member reception. It was great meeting friends from all over the world and sharing their experiences. As I said earlier, I had to crash in the bed at the end.

Keynote by Milton Chang- Straight talk about entrepreneurship

Hello folks,

FiO has been eventful from the very first day – the highlight of the morning for me was the keynote of Milton Chang at student chapter leadership meeting and the meeting itself. The afternoon was made interesting by the ‘Hot topics in Optics’ overview. In this post, I am recounting some of the inspiring thoughts that Milton shared with us during his talk and a long Q&A session that followed.

Apart from an excellent record of successful startups, Milton is known to be a person enthusiastic about sharing his hard-earned life experiences. Today’s talk was a case in point. He used his entrepreneurial milestones of life to describe what approaches succeeded and what did not for him. His talk was followed by an intense and informative question-answer session which was followed my personal 5 minute chat with him. It seems that the video of the talk and Q&A session is not recorded. If you find out otherwise, please post a link to the video in the comment. The key points that he made about succeeding as a tech-startup were :

• Start small and develop slowly but steadily: Milton made a point that a typical VC receives 1000 applications every year, out of which 100 are funded. Out of that around 30% remain viable, which is a success ratio of dismal 0.3%! One of the reasons is that they try to become a big company and put all pieces of infrastructure together at a rapid pace. It pays to remain small until it is required to become big. So his strategy is to grow a startup as the product line and its sales volume grows. In some cases, it implies that you bootstrap the startup by using the returns earned only from that startup. The bootstrapping ensures that you do not end up spending too much if the idea does not work out. He mentioned how Newport took 17 years to become a public company since its inception.
• Focus on your core abilities and provide the best product: Unless you excel at the product and are able to distinguish it in the market from the competition with a definite advantage, it won’t sell. Therefore, his advice was to identify your core abilities and the product’s core advantages at the earliest and amplify them. It is important to have a small success early to be able to distinguish yourself from the wannabes. This small success is usually followed by more exciting and rewarding opportunities.
• Create value in an efficient way: Bootstrapping and focusing on your core abilities allow you to ‘create value efficiently,’ the key ingredient of a viable and happy business. It allows your company to become and remain excellent at what it does.
• Keep up with business literature: After his doctoral education and work in aerospace industry, Milton was called upon by his friend John Matthews to join then newly born Newport.  He was the seventh employee and had just made transition from technology to marketing.  He recounted how he understood nuances of business by reading business literature such as Wall street journal voraciously and cutting out adverts that he thought should be emulated. This allowed him to perceive the direction of the market and create some of the most captivating adverts. This marketing aspect of the business is as important as development of a great product and should receive good part of your attention.
• Have humility of learning from mistakes and asking help: He pointed out how business is benefited by developing trust and deep relationships with your colleagues. If one has humility of knowing limits of his/her abilities and asking for help when required, he/she will have more peaceful nights and better chances of success.

Of course, it is the best to hear the ideas from the source himself.

Till the next post, ciao.

Before the start of Frontiers in Optics – travel and lab visits

Before the conference started, I meant to post one more discussion about phase-space optics describing and simulating the notion of instantaneous frequency. But I will have to wait till my laptop agrees to the plan. I use LaTeX2WP (WP=WordPress) python script by Luca Trevisan for posting mathematical content, and somehow python refuses to get going on my laptop.  Apart from that minuscule disappointment, my time so far in the US has been fantastic!

I have been in the US since October 6 and done good deal of traveling. It is slightly unusual to travel before the conference but I need to return home just after the conference. I am recounting some of the exciting sights and university visits that I did. Hope someone traveling after the conference finds travel information useful.

Let’s go in reverse chronological order. I have  just returned half an hour earlier from a trip to lake Tahoe (right now it is 1.12 am of Oct 11). My friend working in the bay area drove me to this beautiful lake. You may already know that it is on the border of California and Nevada and is one of the most beautiful sights in the US. I learned that the emerald bay of the lake Tahoe is the second most photographed sight. It is a great place to enjoy water-scooter (which is called jetski here), motor boat, and para-sailing among other things. We reached there late in afternoon and wind had picked up (it is end of the summer season). So we were advised against para-sailing and jetski, but had great fun driving a motor-boat for an hour. If you plan to visit Tahoe, check with the people at rental business about the weather and what is possible. It is around 3.5 hours drive from San Jose/San Francisco and if you reach there before lunch, there are better chances of your being able to get more hands-on water experience than driving around a motor-boat.

Before Tahoe, I had the staple diet of tourists in San Francisco : Alcatraz prison, Golden gate bridge, Fisherman’s wharf, Segway tour, Exploratorium and Tactile dome. The first three sites I mentioned need no further mention.

I experienced the `personal transporter’ segway for the first time and I must say it is fantastic. The machine is designed to maintain balance – therefore if you lean forward, it goes forward to regain balance. This balance feedback loop is implemented very well and the machine seems to perceive driver’s intentions.

Tactile dome and the exploratorium were great fun too. Tactile dome is one of the exhibits of the exploratorium. It is a pitch-dark dome which you navigate using only your sense of touch. You pay 17 bucks to enter it (this fee includes entrance to the exploratorium). You have an hour to enter and exit this labyrinth as many times as you like. I found that if you are not scared of dark and just following your senses, you can enter and exit in 15 minutes flat. However, it is amazing that every time you go through it, you discover different structures inside it. I made two trips through the dome  and on my second trip, I discovered a place inside to seat and relax!

I had good brainstorming sessions at Stanford and UCSF. I visited Marc Levoy and Zhengyun Zhang at Stanford to take a peek at their light-field microscope. I found that they have another interesting project going on. It is dubbed Camera 2.0 and it is about designing an open-source camera with programmable hardware and software. Marc is presenting a talk about light-field microscope on Wednesday morning at FiO. We had a great discussion on phase-space representation of imaging systems (topic of my talk on Monday) and registration of gradient information measured through various methods (topic of my talk on Wednesday). Lab visits are great because they can add a lot of new perspective to your repertoire.

At UCSF, I was visiting Nico Stuurman and Arthur Edelstein at Ron Vale’s lab. My key reason for visiting them is their open-source acquisition software micro-manager. I have found this software useful for being able to construct useful automation for custom imaging systems and have contributed to a hardware driver for Olympus scopes. They are an enthusiastic bunch of people willing to make a positive impact in the microscopy world, where proprietary and closed acquisition software have failed to meet the needs of microscopy tinkerers. I shall be working with them to add support for the quantitative phase imaging method (AIDPC) – which we have developed.

An important note if you are visiting Stanford and want to catch some beautiful sights of the campus: make sure you DO NOT carry any heavy luggage (even 75% of the size of the typical cabin bag). Stanford has an arbitrary rule which prohibits even medium sized bags at places worth visiting (e.g. Hoover tower, Cantor arts center). That is not bad – what is bad is that they do not have any place for you to store it and even worse, they do not believe in verifying that you are not carrying anything dangerous.

That’s all for now. Tomorrow (sorry, today) morning is going to be an exciting day as the FiO kicks off . I plan to meet Milton Chang (a successful serial entrepreneur & venture capitalist and the keynote speaker at the OSA student chapter leadership meeting) to capture some of his views about entrepreneurship – especially in the area of open-source projects. Then there will be the ‘Hot in Optics’ summary from 4-6 pm at Fairmont. It’s going to be a busy day, so Good night.

Some verbiage

This is a random thought that occurred to me but interesting nevertheless.

I recently saw V for Vendetta. I was (and still am) preparing slides for my talks and on one of the slides, I had lots of words. Looking at the slide afterwards, suddenly the noun ‘verbiage’ came to my mind (some might say, this blog can inspire similar feelings ). From verbiage, I remembered this self-introduction of ‘V’ from the movie.  Then, I recalled another verbiage that used to be my screen-saver:

“Those who are perforce constrained to be domiciled in vitreous structures with patent frangibility should on no account employ petrous formations as projectiles unto others.”

I think I read the above first at one of the vocab competitions at IIM-Ahmedabad and here is its analysis:

• perforce constrained to be domiciled = forced to live in
• vitreous structures = glass buildings
• patent frangibility = easily broken
• petrous formations = stones!
• employ…as projectile = throw!

Simply put: People who live in glass houses should not throw stones!

If you would like to read more, this is the place.

Alright, I am going back to slides.

The way ideas develop

Adam made a solid point on his blog that people are the key reason academia is so much fun. We all know that research results come out only after following a rather circuitous path and often it happens that interaction with an insightful colleague (mostly our adviser) points us to a shortcut. One of the presentation that I am going to give at FiO has such an incidence behind it.

Martin Bastiaans,  who will be presenting a tutorial at the special symposium on phase-space optics at FiO was recently visiting us on invitation from my adviser, Colin. Our OSA student chapter took this opportunity to launch a local optics seminar series and he was our inaugural speaker at the series. We requested him to give a tutorial on phase-space optics and we had nearly 10 hours of insightful and informative discussion over 3 talks. He has been a member of a church choir and that seems to explain his interest in instantaneous frequency (musical score is perhaps the earliest representation where frequency is plotted against time.)

I have been interested in partially coherent imaging (I should write a sensible post about that sometime). Colin recollected having discussion with Martin once (may be in 80s) that there are very intriguing similarities between models for partially coherent imaging and Wigner distribution. Therefore, there should be a connection between them. Well, may be we found it.

I had been looking for this connection for close to 6 months before Martin visited, but I had not paid attention to a generalised class of phase-space representations (called Cohen class of representations). Martin talked to us about his work that uses the Cohen class distribution for processing of coherent signals. I found it interesting and started reading about it.

And then came one of those ‘Aha!’ moments, where we realized the connection between traditionally used partially coherent model and a yet-to-be-explained Cohen class distribution. I was going to present work based on the earlier model, but that presentation has now morphed into the one based on the new model.  The title of the talk has morphed from ‘Transfer-function analysis of partially coherent imaging systems…’ to ‘Phase-space analysis of partially coherent imaging systems…’

An election worth voting for

Here is one election that lets you choose between good and better rather than bad and worse – as is more often the case. Google has initiated a project for finding the ideas that matter the most and ‘that help the most’. It is called

Google invited ideas from the world about the projects that would help the most number of people. Now it is seeking votes (till October 8th) to determine which five themes will be funded (with total of $10 million) and guided to reality. FAQs suggest that once the ideas are selected, applications of interest will be sought from capable organizations. Here you can vote for your favorite theme. Two themes that looked most worthwhile to me are “Enhance science and engineering education” and “Encourage positive media depictions of engineers and scientists.” But since there is only one vote to cast, I chose the one about improving media depictions of engineers and scientists. Why? Because I am involved with science and would like the world to think better of me? Partly yes.

Phase-space optics 2: Instantaneous correlation and pseudo-symbolic computation in MATLAB

As I noted in the previous post about FiO, I wish to share some of the simulations and basics about phase-space signal analysis. While doing simulations, I sometimes find it tricky to interpret analog equations and convert them to a sequence of programmed operations on discrete data. Also, one needs to be careful about not losing information when going from analog to discrete domain. This all-important step is known as A to D (analog to *discrete*) conversion and is responsible for Nyquist and Shannon being so famous.

I rely on two tools for my modeling needs: MATLAB (TM) most of the time and Mathematica (TM) sometimes. Mathematica is excellent for allowing quick visualization of the model due to its ability to work directly with symbols, whereas MATLAB is good for fast and maintainable implementation. It may be possible to write maintainable code in Mathematica but that requires going through steep learning curve, especially for folks who had learned classic ‘C’ as their first language. Is there a way of marrying symbolic ease of Mathematica with programming ease of MATLAB? I seem to have found one using anonymous functions. The code used in this post has been posted here on MATLAB file-exchange, which will be available within a day after moderation.

Let us consider the problem of calculating instantaneous correlation (which is a primary quantity required to compute and understand the Wigner distribution and Ambiguity function). For a signal it is defined as,

We know that the auto-correlation of a function is defined as Thus, we see that instantaneous correlation is in some sense the quantity which when integrated along gives us the auto-correlation function.

As we shall see in upcoming posts, Wigner distribution is the Fourier transform of the instantaneous correlation (Icorr) along the delay () dimension, whereas the Ambiguity function is Icorr’s Fourier transform along the normal () dimension. Therefore, there are two ways of interpreting what the instantaneous correlation is:

1. is the signal which is shifted by and The conjugate of the delayed version () when multiplied with the advanced version () gives us Icorr.
2. is the signal which is first dilated to give us This dilated signal’s conjugate is flipped to give The signal is advanced to give and the signal is delayed to give us

As is often the case – hindsight is 20/20. I could write above paragraph so clearly because I have already gone through simulating the problem with two tools. But things should be presented logically and not chronologically, so I explained the model first. Next are the simulations.

Century old color photographs of the Russian empire

I always thought color photography was a relatively recent invention, perhaps around the first world war. I was amazed to find this digital exhibit on Library of Congress website that shows color photographs that were taken by a photographer to the Tsar (Prokudin-Gorskii), who was also an accomplished chemist. These photos evoke very lively and rich perception of history – the people and places suddenly acquire the details that were not appreciated before. The description of his color camera and how it was put to use are equally fascinating. Subsequent reading of Wikipedia revealed that James Maxwell is the first person in the recorded history to have taken a color photograph.

Phase-space optics 1: Introduction

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.”

Impressive 3D optical illusion

Hello world,

This illusion is interesting if you haven’t seen before. I am noting an explanation at the end, but do tickle your brain a little before reading it.

Our brain thinks that face is always convex. If you change angle of view, you expect certain facial features to become ‘hidden’. So if no features go out of sight it implies that the face is following your angle of view. This dragon’s face is painted on concave surface – so  you can see the whole face over very wide range. Which the brain misinterprets as the movement