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findintconvo (``find integrals of convolutions'') computes the values of

\int_{0}^{T} \frac{\int\limits_{T_1}^{T_2}
\left[ C_{a}(u) \otimes e^{-k_{2}u} \right] du}{T_2 - T_1} \, w_i \, dt
\end{displaymath} (17)

at many different values of k2, for up to three weighting functions wi. Here, T1 and T2 represent the start and end time any particular frame; in concrete terms, the inner integrand is just the average of $C_{a}(u) \otimes e^{-k_{2}u}$ across each frame, and is computed once for each frame. The outer integral then integrates across all frames to arrive at a single value for each weighting function wi and each value of k2.

The function declaration line.
function [int1, int2, int3] = ...
   findintconvo (Ca_even, ts_even, k2_lookup,...
                 midftimes, flengths, w1, w2, w3, progress)
The input arguments are: = 0=
<879>>=-0.25in 0 =<: <879>>
the blood data resampled at an evenly-spaced time domain. Usually, this is as returned by resampleblood, and possibly delay- and dispersion-corrected by correctblood.
the time domain at which Ca_even was resampled.
the list of k2 values at which to evaluate equation [*]. The output will consist of one value per weighting function per value of k2.
the mid-frame times (used to perform the frame-by-frame integration).
the length of each frame (also used in the frame-by-frame integration).
the three weighting functions wi. w1 must be supplied, but if it is an empty matrix, then a weighting function of 1 is assumed. w2 and w3 are not required, and if they are not given, then int2 and int3 (respectively) will not be defined in the output. Generally, the weighting funtions are simple functions of t such as 1, t, or t2.
an optional argument indicating whether progress should be reported (in the form of printing a dot when the weighted integrals for each value of k2 have been computed). This is quite useful, as findintconvo is one of the most time-consuming steps of RCBF analysis.

The output arguments int1, int2, and int3 are simply vectors containing the value of equation 17 for the three different weighting functions; each element of these vectors corresponds to one value of k2 as supplied in k2_lookup.

Do some initial setup. We need to get the sizes of various vectors, and initialize vectors that we will fill element by element later. Initializing vectors to all zero before filling them allows better memory management by MATLAB.
NumEvenTimes = length(ts_even);
NumFrames = length(midftimes);
fstart = midftimes - (flengths / 2);

TableSize = length (k2_lookup);
integrand = zeros (NumFrames, 1);

if (nargin >= 6); int1 = zeros (1, TableSize); end;
if (nargin >= 7); int2 = zeros (1, TableSize); end;
if (nargin == 8); int3 = zeros (1, TableSize); end;

% if w1 is empty, assume that it should be all ones

if isempty (w1)
   w1 = ones (size(NumFrames));

Calculate each element of the integrals, one at a time. Unfortunately, there does not seem to be any way to vectorize this operation, and it must therefore be performed within an inefficient for loop.

for i = 1:TableSize

First calculate the innermost terms of equation 17. exp_fun is e-k2 t, and convo is $C_a (t) \otimes e^{-k_2 t}$. We then perform the inner integration, i.e. average across all frames. This is performed by nframeint, a special CMEX routine that has been carefully optimised for this very common operation in RCBF analysis. Type help nframeints in MATLAB for more information.
   exp_fun = exp(-k2_lookup(i) * ts_even);
   convo = nconv(Ca_even, exp_fun, ts_even(2) - ts_even(1));

   integrand = nframeint (ts_even, convo(1:length(ts_even)),...
                          fstart, flengths);

Find any frames that are not completely covered by the ts_even time domain. nframeint actually does this for us, returning NaN (not-a-number) in any such element of its output vector. Thus, we only use data from frames that are not equal to NaN, selected using the built-in isnan function.
   select = ~isnan(integrand);

Now we calculate the outer integral (the weighted integral across all frames) for each weighting function with the current value of k2. Again, we use a CMEX routine to enhance performance; ntrapz is a replacement for MATLAB's trapz function that is much faster, as well as having an optional argument to allow weighting of the integrand. Note the use of select to make sure that we only use frames actually spanned by the blood data.
  int1 (i) = ntrapz(midftimes(select), integrand(select), w1(select));
  int2 (i) = ntrapz(midftimes(select), integrand(select), w2(select));
  int3 (i) = ntrapz(midftimes(select), integrand(select), w3(select));
(We have omitted the code that makes w2 and w3 optional in favour of concentrating on the numerical analysis.)

End the for loop.

next up previous contents
Next: B_CURVE Up: Annotated Program Listings Previous: CORRECTBLOOD
Mark Wolforth <>
Greg Ward <>
Sean Marrett <>