Lukas.Ahrenberg » Research http://lukas.ahrenberg.se Sun, 08 Jan 2012 12:53:13 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 Sunday morning complexity http://lukas.ahrenberg.se/archives/705 http://lukas.ahrenberg.se/archives/705#comments Sun, 08 Jan 2012 12:53:13 +0000 lukas http://lukas.ahrenberg.se/?p=705 Wondering what to do with your Sunday? I know just the very thing: complexity!

Nature Physics has an insights issue on the subject (vol. 8 no. 1), and I have heard that the articles are free of charge until February 1, 2012 (the Nature journals are unfortunately otherwise quite fond of paywalls). There’s quite a lot of things to read (though I cheated a bit – I got some through early access and did not read it all on [Sun]day).

First, there is a good Commentary by Albert-László Barabási – The network takeover (doi:10.1038/nphys2188) – arguing that while we may not be seeing the end of reductionism, the advent complexity science and network theory is an important part of a new trend where the structure of component relations is studied. I agree, I think more and more of natural science is turning to studying emergent behaviour, putting back together that what has been taken apart, and the rigorous theories developed over the years in mathematics, physics, and information theory are providing new ways in for instance in the  social sciences.

There are also three good reviews in the issue, highlighting what perhaps are the main network science directions currently: the information theoretical view of a system, the analysis of structure in existing networks, and the simulation of dynamic systems as processes on networks.

Between order and chaos by James Crutchfield (doi:10.1038/nphys2190) address randomness and computational mechanics. Very nice review describing complexity from an information theoretical point of view, and therefore having a special place in my heart. Good if you are approaching the field, as I do, from that specific direction and have been telling yourself “Hey, this look awfully lot like computation to me.” It all comes down to \epsilon-machines.

Communities, modules and large-scale structure in networks by M. E. J. Newman (doi:10.1038/nphys2162). Looking at structure, and communities in networks.  Besides being an interesting problem, community detection has turned out to be a quite important problem in many applications (for example lumping you together with your social network friends in order to predict your behaviour — wait you did not think you were unique, did you?). Newman’s review is well written and I felt I knew the field better after reading it. Both the historic link to physics, the challenges, and the state of the research.

Modelling dynamical processes in complex socio-technical systems by Alessandro Vespignani (doi:10.1038/nphys2160) is an overview of what one may gain from viewing a complex system as a process on a network instead of as a compartmental model, and mean field approaches. Borrows the standard SI(R) examples from epidemiology, but it is of course the same process whether we talk about diffusion, epidemics, or memes on twitter.

The Insight issue also contain a progress article – Networks formed from interdependent networks by Gao et al. (doi:10.1038/nphys2180) – but I have to admit not having read it yet. It is more technical dealing with the issue of a network of networks. Something very interesting as I guess it may mean that some of the sub-networks are not any more in equilibrium, but also something I am not familiar with yet.

In any case, good stuff for a Sunday read, go grab the PDFs while you can!

If you still can’t get enough, or simply want a good introduction to the whole complexity thing, Newman has recently published another good general review Complex Systems: A Survey (properly in Am. J. Phys. 79, 800-810 (2011), I believe, but I took the liberty of linking to the arXiv.org preprint).

Have fun!

.L

]]> http://lukas.ahrenberg.se/archives/705/feed 0 Errata/Update to ‘Computer generated holograms from three dimensional meshes using an analytic light transport model’ http://lukas.ahrenberg.se/archives/588 http://lukas.ahrenberg.se/archives/588#comments Wed, 30 Nov 2011 22:16:25 +0000 lukas http://lukas.ahrenberg.se/?p=588 Section 3.B; Equations 15 – 19

A few weeks ago I received an email asking about the article Computer generated holograms from three-dimensional meshes using an analytic light transport model by myself and three colleagues published in Applied Optics back in 2008. After taking a second look at the section in question ( 3.B) I had to agree, a couple of the steps were hard to follow, and Eqn. 16 seemed especially confusing.  So, let me correct this by pointing out how to get from Eqn. 15  to Eqn. 19. in some more detail than the paper allows for.

First however, let me correct some actual errors I spotted when looking at this section.

  1. Eqn. 16. The vectors \begin{bmatrix} x \\ y \end{bmatrix} and \begin{bmatrix} s \\ t \end{bmatrix} should change place. In addition it is perhaps not obvious how to get from Eqn. 15 to Eqn. 16; see below for a (hopefully) clearer derivation.
  2. \mathbf{J} at this point in the text is not strictly a Jacobian determinant I guess. Just a plain old determinant. The Jacobian one comes in later for the change of coordinates. Writing error there.
  3. The switch of s,t and x,y coordinates seems to have crept in to Eqn. 18 as well.
  4. Another typo: Eqn. 19 – the first argument for F_{\Delta} should be \frac{(a_{22}u - a_{21}v)}{J}.

Expanded explanation

Now, let me take you from roughly Eqn. 15 to Eqn. 19. We have two ‘triangle functions’ (defined earlier in the text so I’ll be brief here) in the plane: f_{\Delta} with vertices \left(0,0\right), \left(1,0\right), \left(1,1\right) (The triangle \Delta); and f_{\Gamma} with vertices \left(s_1, t_1\right), \left(s_2, t_2\right), \left(s_3, t_3\right) (The triangle \Gamma).

Moreover we have an expression for the  Fourier transform of f_{\Delta}, denoted F_{\Delta} (it is given earlier in the article and rather  long, so I’ll just use a symbol for it here), and now we would like to know if we can express the Fourier transform of f_{\Gamma}, denoted F_{\Gamma}  as a function of this.

Turns out we can; reference 16 in the paper: ‘Affine theorem for two-dimensional fourier transform’ by Bracewell et al. is a short note showing how an affine transform relating the domains of two functions can be used to relate the corresponding Fourier spectra.

Now,  as \Delta and \Gamma are two triangles it is clear that we can set up an affine transform relating them. This is the coordinate transform in Eqn 15:

\begin{bmatrix} s \\ t \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a_{13} \\ a_{23} \end{bmatrix} .

Here, the xy-vector represents the f_{\Delta} coordinates and the st-vector the f_{\Gamma} coordinates.  Finding the elements a_{ij} is straight forward. If we define the following relation between the vertices of the two triangles: \left(0, 0\right) \mapsto \left(s_1, t_1\right), \left(1, 0\right) \mapsto \left(s_2, t_2\right), \left(1, 1\right) \mapsto \left(s_3, t_3\right) we can set up the following relations using Eqn 15:

\begin{bmatrix} s_1 \\ t_1 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} + \begin{bmatrix} a_{13} \\ a_{23} \end{bmatrix}

\begin{bmatrix} s_2 \\ t_2 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} a_{13} \\ a_{23} \end{bmatrix}

\begin{bmatrix} s_3 \\ t_3 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} + \begin{bmatrix} a_{13} \\ a_{23} \end{bmatrix}

Performing the matrix calculations leads to the following system of equations:

\begin{bmatrix} s_1 \\ t_1 \end{bmatrix} = \begin{bmatrix} a_{13} \\ a_{23} \end{bmatrix}

\begin{bmatrix} s_2 \\ t_2 \end{bmatrix} = \begin{bmatrix} a_{11} + a_{13} \\ a_{21} + a_{23} \end{bmatrix}

\begin{bmatrix} s_3 \\ t_3 \end{bmatrix} = \begin{bmatrix} a_{11} + a_{12} + a_{13} \\ a_{21} + a_{22} +a_{23} \end{bmatrix}

Which solved for a_{ij} gives the following transform (Eqn. 15 witha_{ij} filled in)

\begin{bmatrix} s \\ t \end{bmatrix} = \begin{bmatrix} s_2 - s_1 & s_3 - s_2\\ t_2 - t_1 & t_3 - t_2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} s_1 \\ t_1 \end{bmatrix} .

This transform maps from triangle \Delta to triangle \Gamma.  However, using the technique of Bracewell et al. requires a transform from the domain of f_{\Gamma} to f_{\Delta} because we want to take evaluate the former function as a look-up using the latter. That is, we need to map from st-coordinates to xy-coordinates, which is the inverse of the affine transform in Eqn 15.

Calculating the inverse is straight forward. If the original system is expressed as

\begin{bmatrix} s \\t \end{bmatrix} = \mathbf{A} \begin{bmatrix} x\\y \end{bmatrix} + \mathbf{a},

where \mathbf{A} is the 2×2 matrix of Eqn. 15, and \mathbf{a} is the 2×1 vector,

then its inverse is a new affine transform

\begin{bmatrix} x \\ y \end{bmatrix} = \mathbf{B} \begin{bmatrix} s\\t \end{bmatrix} + \mathbf{b} ;

where \mathbf{B} = \mathbf{A}^{-1} and \mathbf{b} = - \mathbf{A}^{-1}\mathbf{a}.

Using the values for a_{ij} calculated above give the transform

\begin{bmatrix} x \\ y\end{bmatrix} = \frac{1}{J}\begin{bmatrix}t_3 - t_2 & s_2 - s_3\\ t_1 - t_2 & s_2 - s_1 \end{bmatrix} \begin{bmatrix} s \\ t \end{bmatrix} - \frac{1}{J} \begin{bmatrix} s_1 (t_3 - t_2)+t_1 (s_2 - s_3) \\ s_1 (t_1 - t_2)+t_1 ( s_2 - s_1) \end{bmatrix}

where J = \det{\mathbf{B}} = (t_3 - t_2)( s_2 - s_1) - (s_2 - s_3)(t_1 - t_2).  (The determinant of \mathbf{B} which I still chose to call J to stay with the notation of the paper.)

The above are then the corrected versions of Eqns. 16 – 17.

From here it is quite straight forward to plug the result directly int0 the expression from Bracewell et al. to yield Eqn. 18 and Eqn. 19 of the paper, however because the notation is not very clear I will explain it further.

Our paper uses a_{ij} to denote the matrix elements used in Eqns. 18-19. These are meant to refer to the elements of the matrix in Eqn. 16, however since a_{ij} is used in Eqn. 15, this is quite confusing. Following the notation in this post, the elements should rather be called b_{ij}; with Eqn. 18  expressed as (while also correcting the switch of xy/st coordinates):

f_{\Gamma}\left(s,t\right) = f_{\Delta}\left(b_{11} s + b_{12}t + b_{13}, b_{11} s + b_{12}t + b_{13}\right).

Eqn. 19 then, does not change (except that we now use b_{ij}, and the correction of the misprinted index):

F_{\Gamma}\left(u,v\right) = \frac{1}{|J|}\exp\left\{\frac{2\pi i}{J}\left[\left(b_{22}b_{13}-b_{12}b_{23}\right)u+\left(b_{11}b_{23}-b_{13}b_{21}\right)v\right]\right\}\\ \; \; \times F_{\Delta}\left(\frac{1}{J}\left(b_{22}u-b_{21}v\right),\frac{1}{J}\left(-b_{12}u+b_{11}v\right)\right).

J = \det B as defined above, and b_{ij} taken from Eqn. 16.

I hope this extended explanation/derivation explain things and will help future readers of the paper.

Thanks to Zhang Jianquang for asking me about the mathematics and thus prompting me to take a second look.

Please let me know if you find any errors in this post, or in the paper itself and I will try to correct them.

When I have time I think I will check through the rest of the paper. When writing this I had a thought regarding the generated wave field and I think that a diffusing field may be needed as well.

.L

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