... quality).1
Further, hardware implementations of algorithms often yield speed increases of approximately an order of magnitude. This algorithm may not be suited to having a full hardware implementation (as it requires substantial amounts of memory for reasonable resolution images), but parts of it may be suited to hardware implementations. 
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... dimensions2
Which also has the additional feature of being its own Fourier transform, that is, \( \mathbf{F}(f)=f \) for \( f(\vec{x})=e^{-\left\vert \vec{x}\right\vert ^{2}} \). More generally, the Fourier transform of a Gaussian function is also a Gaussian function (with different constants). 
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... shells3
For the three dimensional images that we are using. Naturally if we were dealing with two dimensional images we would expect to find thin rings. 
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...\( r \)4
Or at the very least, a mask-based smoothing operation as detailed in Section 2.1.1. 
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... integer)5
Implemented by taking the first 100 terms of the infinite sum describing \( J_{n} \), and precomputing the values of \( J_{n} \) for the values of \( z \) that would be required. The sum used was

\begin{displaymath} J_{n}(x)=x^{n}\sum _{m=0}^{\infty }\frac{(-1)^{m}x^{2m}}{2^{2m+n}m!(n+m)!}\end{displaymath}

as derived in [Kreyszig, 1993, Section 5.5, page 227]. 
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... values6
The file format actually has an option to use (big-endian) word length pixels (giving 65536 possible pixel values), however none of our data files used this feature. 
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... 256)7
This was also observed to be a natural choice, as the images produced by the equipment have low level noise values of approximately this value where no signal is received. 
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... points8
Here, 1k = 1024, as is standard for computing applications. 
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... time9
Note that this is the CPU time required to do the calculations. The actual wall-clock time taken for the program to run was substantially higher than these figures, due to the large amounts of IO required (and often also virtual memory swapping). The machine used to run these tests was an Intel Celeron 300Mhz (128kb CPU cache, overclocked to 450MHz) with 128Mb RAM running Linux. 
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... time10
Here we are using \( \Theta \)-notation (related to big-Oh notation), which denotes the set of functions \( \left\{ f(n)\right\} \) that are asymptotically bound by \( g(n) \)

\begin{displaymath} \Theta (g(n))=\left\{ f(n): \exists c_{1}>0, c_{2}>0,... ...c_{1}g(n)\leq f(n)\leq c_{2}g(n) \forall n\geq n_{0}\right\} \end{displaymath}

Also, in our notation \( \lg x=\log _{2}x \). 

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... well11
Compared to the exact solutions which have exponential time complexity. 
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