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Errata/Update to ‘Computer generated holograms from three dimensional meshes using an analytic light transport model’

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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