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I have been attempting to detect peaks in sinusoidal time-series data in real time, however I've had no success thus far. I cannot seem to find a real-time algorithm that works to detect peaks in sinusoidal signals with a reasonable level of accuracy. I either get no peaks detected, or I get a zillion points along the sine wave being detected as peaks.
What is a good real-time algorithm for input signals that resemble a sine wave, and may contain some random noise?
As a simple test case, consider a stationary, sine wave that is always the same frequency and amplitude. (The exact frequency and amplitude don't matter; I have arbitrarily chosen a frequency of 60 Hz, an amplitude of +/− 1 unit, at a sampling rate of 8 KS/s.) The following MATLAB code will generate such a sinusoidal signal:
dt = 1/8000;
t = (0:dt:(1-dt)/4)';
x = sin(2*pi*60*t);
Using the algorithm developed and published by Jean-Paul, I either get no peaks detected (left) or a zillion "peaks" detected (right):
I've tried just about every combination of values for these 3 parameters that I could think of, following the "rules of thumb" that Jean-Paul gives, but I have so far been unable to get my expected result.
I found an alternative algorithm, developed and published by Eli Billauer, that does give me the results that I want—e.g.:
Even though Eli Billauer's algorithm is much simpler and does tend to reliably produce the results that I want, it is not suitable for real-time applications.
As another example of a signal that I'd like to apply such an algorithm to, consider the test case given by Eli Billauer for his own algorithm:
t = 0:0.001:10;
x = 0.3*sin(t) + sin(1.3*t) + 0.9*sin(4.2*t) + 0.02*randn(1, 10001);
This is a more unusual (less uniform/regular) signal, with a varying frequency and amplitude, but still generally sinusoidal. The peaks are plainly obvious to the eye when plotted, but hard to identify with an algorithm.
What is a good real-time algorithm to correctly identify the peaks in a sinusoidal input signal? I am not really an expert when it comes to signal processing, so it would be helpful to get some rules of thumb that consider sinusoidal inputs. Or, perhaps I need to modify e.g. Jean-Paul's algorithm itself in order to work properly on sinusoidal signals. If that's the case, what modifications would be required, and how would I go about making these?
Case 1: sinusoid without noise
If your sinusoid does not contain any noise, you can use a very classic signal processing technique: taking the first derivative and detecting when it is equal to zero.
For example:
function signal = derivesignal( d )
% Identify signal
signal = zeros(size(d));
for i=2:length(d)
if d(i-1) > 0 && d(i) <= 0
signal(i) = +1; % peak detected
elseif d(i-1) < 0 && d(i) >= 0
signal(i) = -1; % trough detected
end
end
end
Using your example data:
% Generate data
dt = 1/8000;
t = (0:dt:(1-dt)/4)';
y = sin(2*pi*60*t);
% Add some trends
y(1:1000) = y(1:1000) + 0.001*(1:1000)';
y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';
% Approximate first derivative (delta y / delta x)
d = [0; diff(y)];
% Identify signal
signal = derivesignal(d);
% Plot result
figure(1); clf; set(gcf,'Position',[0 0 677 600])
subplot(4,1,1); hold on;
title('Data');
plot(t,y);
subplot(4,1,2); hold on;
title('First derivative');
area(d);
ylim([-0.05, 0.05]);
subplot(4,1,3); hold on;
title('Signal (-1 for trough, +1 for peak)');
plot(t,signal); ylim([-1.5 1.5]);
subplot(4,1,4); hold on;
title('Signals marked on data');
markers = abs(signal) > 0;
plot(t,y); scatter(t(markers),y(markers),30,'or','MarkerFaceColor','red');
This yields:
This method will work extremely well for any type of sinusoid, with the only requirement that the input signal contains no noise.
Case 2: sinusoid with noise
As soon as your input signal contains noise, the derivative method will fail. For example:
% Generate data
dt = 1/8000;
t = (0:dt:(1-dt)/4)';
y = sin(2*pi*60*t);
% Add some trends
y(1:1000) = y(1:1000) + 0.001*(1:1000)';
y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';
% Add some noise
y = y + 0.2.*randn(2000,1);
Will now generate this result because first differences amplify noise:
Now there are many ways to deal with noise, and the most standard way is to apply a moving average filter. One disadvantage of moving averages is that they are slow to adapt to new information, such that signals may be identified after they have occurred (moving averages have a lag).
Another very typical approach is to use Fourier Analysis to identify all the frequencies in your input data, disregard all low-amplitude and high-frequency sinusoids, and use the remaining sinusoid as a filter. The remaining sinusoid will be (largely) cleansed from the noise and you can then use first-differencing again to determine the peaks and troughs (or for a single sine wave you know the peaks and troughs happen at 1/4 and 3/4 pi of the phase). I suggest you pick up any signal processing theory book to learn more about this technique. Matlab also has some educational material about this.
If you want to use this algorithm in hardware, I would suggest you also take a look at WFLC (Weighted Fourier Linear Combiner) with e.g. 1 oscillator or PLL (Phase-Locked Loop) that can estimate the phase of a noisy wave without doing a full Fast Fourier Transform. You can find a Matlab algorithm for a phase-locked loop on Wikipedia.
I will suggest a slightly more sophisticated approach here that will identify the peaks and troughs in real-time: fitting a sine wave function to your data using moving least squares minimization with initial estimates from Fourier analysis.
Here is my function to do that:
function [result, peaks, troughs] = fitsine(y, t, eps)
% Fast fourier-transform
f = fft(y);
l = length(y);
p2 = abs(f/l);
p1 = p2(1:ceil(l/2+1));
p1(2:end-1) = 2*p1(2:end-1);
freq = (1/mean(diff(t)))*(0:ceil(l/2))/l;
% Find maximum amplitude and frequency
maxPeak = p1 == max(p1(2:end)); % disregard 0 frequency!
maxAmplitude = p1(maxPeak); % find maximum amplitude
maxFrequency = freq(maxPeak); % find maximum frequency
% Initialize guesses
p = [];
p(1) = mean(y); % vertical shift
p(2) = maxAmplitude; % amplitude estimate
p(3) = maxFrequency; % phase estimate
p(4) = 0; % phase shift (no guess)
p(5) = 0; % trend (no guess)
% Create model
f = #(p) p(1) + p(2)*sin( p(3)*2*pi*t+p(4) ) + p(5)*t;
ferror = #(p) sum((f(p) - y).^2);
% Nonlinear least squares
% If you have the Optimization toolbox, use [lsqcurvefit] instead!
options = optimset('MaxFunEvals',50000,'MaxIter',50000,'TolFun',1e-25);
[param,fval,exitflag,output] = fminsearch(ferror,p,options);
% Calculate result
result = f(param);
% Find peaks
peaks = abs(sin(param(3)*2*pi*t+param(4)) - 1) < eps;
% Find troughs
troughs = abs(sin(param(3)*2*pi*t+param(4)) + 1) < eps;
end
As you can see, I first perform a Fourier transform to find initial estimates of the amplitude and frequency of the data. I then fit a sinusoid to the data using the model a + b sin(ct + d) + et. The fitted values represent a sine wave of which I know that +1 and -1 are the peaks and troughs, respectively. I can therefore identify these values as the signals.
This works very well for sinusoids with (slowly changing) trends and general (white) noise:
% Generate data
dt = 1/8000;
t = (0:dt:(1-dt)/4)';
y = sin(2*pi*60*t);
% Add some trends
y(1:1000) = y(1:1000) + 0.001*(1:1000)';
y(1001:2000) = y(1001:2000) - 0.002*(1:1000)';
% Add some noise
y = y + 0.2.*randn(2000,1);
% Loop through data (moving window) and fit sine wave
window = 250; % How many data points to consider
interval = 10; % How often to estimate
result = nan(size(y));
signal = zeros(size(y));
for i = window+1:interval:length(y)
data = y(i-window:i); % Get data window
period = t(i-window:i); % Get time window
[output, peaks, troughs] = fitsine(data,period,0.01);
result(i-interval:i) = output(end-interval:end);
signal(i-interval:i) = peaks(end-interval:end) - troughs(end-interval:end);
end
% Plot result
figure(1); clf; set(gcf,'Position',[0 0 677 600])
subplot(4,1,1); hold on;
title('Data');
plot(t,y); xlim([0 max(t)]); ylim([-4 4]);
subplot(4,1,2); hold on;
title('Model fit');
plot(t,result,'-k'); xlim([0 max(t)]); ylim([-4 4]);
subplot(4,1,3); hold on;
title('Signal (-1 for trough, +1 for peak)');
plot(t,signal,'r','LineWidth',2); ylim([-1.5 1.5]);
subplot(4,1,4); hold on;
title('Signals marked on data');
markers = abs(signal) > 0;
plot(t,y,'-','Color',[0.1 0.1 0.1]);
scatter(t(markers),result(markers),30,'or','MarkerFaceColor','red');
xlim([0 max(t)]); ylim([-4 4]);
Main advantages of this approach are:
You have an actual model of your data, so you can predict signals in the future before they happen! (e.g. fix the model and calculate the result by inputting future time periods)
You don't need to estimate the model every period (see parameter interval in the code)
The disadvantage is that you need to select a lookback window, but you will have this problem with any method that you use for real-time detection.
Video demonstration
Data is the input data, Model fit is the fitted sine wave to the data (see code), Signal indicates the peaks and troughs and Signals marked on data gives an impression of how accurate the algorithm is. Note: watch the model fit adjust itself to the trend in the middle of the graph!
That should get you started. There are also a lot of excellent books on signal detection theory (just google that term), which will go much further into these types of techniques. Good luck!
Consider using findpeaks, it is fast, which may be important for realtime. You should filter high-frequency noise to improve accuracy. here I smooth the data with a moving window.
t = 0:0.001:10;
x = 0.3*sin(t) + sin(1.3*t) + 0.9*sin(4.2*t) + 0.02*randn(1, 10001);
[~,iPeak0] = findpeaks(movmean(x,100),'MinPeakProminence',0.5);
You can time the process (0.0015sec)
f0 = #() findpeaks(movmean(x,100),'MinPeakProminence',0.5)
disp(timeit(f0,2))
To compare, processing the slope is only a bit faster (0.00013sec), but findpeaks have many useful options, such as minimum interval between peaks etc.
iPeaks1 = derivePeaks(x);
f1 = #() derivePeaks(x)
disp(timeit(f1,1))
Where derivePeaks is:
function iPeak1 = derivePeaks(x)
xSmooth = movmean(x,100);
goingUp = find(diff(movmean(xSmooth,100)) > 0);
iPeak1 = unique(goingUp([1,find(diff(goingUp) > 100),end]));
iPeak1(iPeak1 == 1 | iPeak1 == length(iPeak1)) = [];
end
I am trying to implement the gradient descent algorithm to fit a straight line to noisy data following the following image taken from Andrew Ng's course.
First, I am declaring the noisy straight line I want to fit:
xrange =(-10:0.1:10); % data lenght
ydata = 2*(xrange)+5; % data with gradient 2, intercept 5
plot(xrange,ydata); grid on;
noise = (2*randn(1,length(xrange))); % generating noise
target = ydata + noise; % adding noise to data
figure; scatter(xrange,target); grid on; hold on; % plot a sctter
I then initialize both parameters, the objective function history as follows:
tita0 = 0 %intercept (randomised)
tita1 = 0 %gradient (randomised)
% Initialize Objective Function History
J_history = zeros(num_iters, 1);
% Number of training examples
m = (length(xrange));
I proceed to write the gradient descent algorithm:
for iter = 1:num_iters
h = tita0 + tita1.*xrange; % building the estimated
%c = (1/(2*length(xrange)))*sum((h-target).^2)
temp0 = tita0 - alpha*((1/m)*sum((h-target)));
temp1 = tita1 - alpha*((1/m)*sum((h-target))).*xrange;
tita0 = temp0;
tita1 = temp1;
J_history(iter) = (1/(2*m))*sum((h-target).^2); % Calculating cost from data to estimate
end
Last but not least, the plots. I am using MATLAB's inbuilt polyfit function to test the accuracy of my fit.
% print theta to screen
fprintf('Theta found by gradient descent: %f %f\n',tita0, tita1(end));
fprintf('Minimum of objective function is %f \n',J_history(num_iters));
%Plot the linear fit
hold on; % keep previous plot visibledesg
plot(xrange, tita0+xrange*tita1(end), '-'); title(sprintf('Cost is %g',J_history(num_iters))); % plotting line on scatter
% Validate with polyfit fnc
poly_theta = polyfit(xrange,ydata,1);
plot(xrange, poly_theta(1)*xrange+poly_theta(2), 'y--');
legend('Training data', 'Linear regression','Linear regression with polyfit')
hold off
Result:
AS can be seen my linear regression is not working well at all. It seems as though both parameters (y-intercept and gradient) are not converging to the optimal solution.
Any suggestions on what I may be doing wrong in my implementation would be appreciated. I can't seem to understand where my solution is diverging from the equations shown above. Thanks!
Your implementation for theta_1 is incorrect. Andrew Ng's equation sums across the x's as well. What you have for theta_0 and theta_1 are
temp0 = tita0 - alpha*((1/m)*sum((h-target)));
temp1 = tita1 - alpha*((1/m)*sum((h-target))).*xrange;
Notice that sum((h-target)) appears in both formulas. You will need to multiply the x's first before summing it up. I am not a MatLab programmer so I can't fix your code.
The big picture of what you are doing in your incorrect implementation is that you are pushing the predicted values for the intercept and slope in the same direction because your change is always proportional to sum((h-target)). That's not the way that gradient descent works.
Change your update rule for tita1 as follows:
temp1 = tita1 - alpha*((1/m)*sum((h-target).*xrange));
Also, another remark is you don't really need the temporary variable.
By setting
num_iters = 100000
alpha = 0.001
I can recover
octave:152> tita0
tita0 = 5.0824
octave:153> tita1
tita1 = 2.0085
I have a matrix named figmat from which I obtain the following pcolor plot (Matlab-Version R 2016b).
Basically I only want to extract the bottom red high intensity line from this plot.
I thought of doing it in some way of extracting the maximum values from the matrix and creating some sort of mask on the main matrix. But I'm not understanding a possible way to achieve this. Can it be accomplished with the help of any edge/image detection algorithms?
I was trying something like this with the following code to create a mask
A=max(figmat);
figmat(figmat~=A)=0;
imagesc(figmat);
But this gives only the boundary of maximum values. I also need the entire red color band.
Okay, I assume that the red line is linear and its values can uniquely be separated from the rest of the picture. Let's generate some test data...
[x,y] = meshgrid(-5:.2:5, -5:.2:5);
n = size(x,1)*size(x,2);
z = -0.2*(y-(0.2*x+1)).^2 + 5 + randn(size(x))*0.1;
figure
surf(x,y,z);
This script generates a surface function. Its set of maximum values (x,y) can be described by a linear function y = 0.2*x+1. I added a bit of noise to it to make it a bit more realistic.
We now select all points where z is smaller than, let's say, 95 % of the maximum value. Therefore find can be used. Later, we want to use one-dimensional data, so we reshape everything.
thresh = min(min(z)) + (max(max(z))-min(min(z)))*0.95;
mask = reshape(z > thresh,1,n);
idx = find(mask>0);
xvec = reshape(x,1,n);
yvec = reshape(y,1,n);
xvec and yvec now contain the coordinates of all values > thresh.
The last step is to do some linear polynomial over all points.
pp = polyfit(xvec(idx),yvec(idx),1)
pp =
0.1946 1.0134
Obviously these are roughly the coefficients of y = 0.2*x+1 as it should be.
I do not know, if this also works with your data, since I made some assumptions. The threshold level must be chosen carefully. Maybe some preprocessing must be done to dynamically detect this level if you really want to process your images automatically. There might also be a simpler way to do it... but for me this one was straight forward without the need of any toolboxes.
By assuming:
There is only one band to extract.
It always has the maximum values.
It is linear.
I can adopt my previous answer to this case as well, with few minor changes:
First, we get the distribution of the values in the matrix and look for a population in the top values, that can be distinguished from the smaller values. This is done by finding the maximum value x(i) on the histogram that:
Is a local maximum (its bin is higher than that of x(i+1) and x(i-1))
Has more values above it than within it (the sum of the height of bins x(i+1) to x(end) < the height of bin x):
This is how it is done:
[h,x] = histcounts(figmat); % get the distribution of intesities
d = diff(fliplr(h)); % The diffrence in bin height from large x to small x
band_min_ind = find(cumsum(d)>size(figmat,2) & d<0, 1); % 1st bin that fit the conditions
flp_val = fliplr(x); % the value of x from large to small
band_min = flp_val(band_min_ind); % the value of x that fit the conditions
Now we continue as before. Mask all the unwanted values, interpolate the linear line:
mA = figmat>band_min; % mask all values below the top value mode
[y1,x1] = find(mA,1); % find the first nonzero row
[y2,x2] = find(mA,1,'last'); % find the last nonzero row
m = (y1-y2)/(x1-x2); % the line slope
n = y1-m*x1; % the intercept
f_line = #(x) m.*x+n; % the line function
And if we plot it we can see the red line where the band for detection was:
Next, we can make this line thicker for a better representation of this line:
thick = max(sum(mA)); % mode thickness of the line
tmp = (1:thick)-ceil(thick/2); % helper vector for expanding
rows = bsxfun(#plus,tmp.',floor(f_line(1:size(A,2)))); % all the rows for each column
rows(rows<1) = 1; % make sure to not get out of range
rows(rows>size(A,1)) = size(A,1); % make sure to not get out of range
inds = sub2ind(size(A),rows,repmat(1:size(A,2),thick,1)); % convert to linear indecies
mA(inds) = true; % add the interpolation to the mask
result = figmat.*mA; % apply the mask on figmat
Finally, we can plot that result after masking, excluding the unwanted areas:
imagesc(result(any(result,2),:))
I have an image of size as RGB uint8(576,720,3) where I want to classify each pixel to a set of colors. I have transformed using rgb2lab from RGB to LAB space, and then removed the L layer so it is now a double(576,720,2) consisting of AB.
Now, I want to classify this to some colors that I have trained on another image, and calculated their respective AB-representations as:
Cluster 1: -17.7903 -13.1170
Cluster 2: -30.1957 40.3520
Cluster 3: -4.4608 47.2543
Cluster 4: 46.3738 36.5225
Cluster 5: 43.3134 -17.6443
Cluster 6: -0.9003 1.4042
Cluster 7: 7.3884 11.5584
Now, in order to classify/label each pixel to a cluster 1-7, I currently do the following (pseudo-code):
clusters;
for each x
for each y
ab = im(x,y,2:3);
dist = norm(ab - clusters); // norm of dist between ab and each cluster
[~, idx] = min(dist);
end
end
However, this is terribly slow (52 seconds) because of the image resolution and that I manually loop through each x and y.
Are there some built-in functions I can use that performs the same job? There must be.
To summarize: I need a classification method that classifies pixel images to an already defined set of clusters.
Approach #1
For a N x 2 sized points/pixels array, you can avoid permute as suggested in the other solution by Luis, which could slow down things a bit, to have a kind of "permute-unrolled" version of it and also let's bsxfun work towards a 2D array instead of a 3D array, which must be better with performance.
Thus, assuming clusters to be ordered as a N x 2 sized array, you may try this other bsxfun based approach -
%// Get a's and b's
im_a = im(:,:,2);
im_b = im(:,:,3);
%// Get the minimum indices that correspond to the cluster IDs
[~,idx] = min(bsxfun(#minus,im_a(:),clusters(:,1).').^2 + ...
bsxfun(#minus,im_b(:),clusters(:,2).').^2,[],2);
idx = reshape(idx,size(im,1),[]);
Approach #2
You can try out another approach that leverages fast matrix multiplication in MATLAB and is based on this smart solution -
d = 2; %// dimension of the problem size
im23 = reshape(im(:,:,2:3),[],2);
numA = size(im23,1);
numB = size(clusters,1);
A_ext = zeros(numA,3*d);
B_ext = zeros(numB,3*d);
for id = 1:d
A_ext(:,3*id-2:3*id) = [ones(numA,1), -2*im23(:,id), im23(:,id).^2 ];
B_ext(:,3*id-2:3*id) = [clusters(:,id).^2 , clusters(:,id), ones(numB,1)];
end
[~, idx] = min(A_ext * B_ext',[],2); %//'
idx = reshape(idx, size(im,1),[]); %// Desired IDs
What’s going on with the matrix multiplication based distance matrix calculation?
Let us consider two matrices A and B between whom we want to calculate the distance matrix. For the sake of an easier explanation that follows next, let us consider A as 3 x 2 and B as 4 x 2 sized arrays, thus indicating that we are working with X-Y points. If we had A as N x 3 and B as M x 3 sized arrays, then those would be X-Y-Z points.
Now, if we have to manually calculate the first element of the square of distance matrix, it would look like this –
first_element = ( A(1,1) – B(1,1) )^2 + ( A(1,2) – B(1,2) )^2
which would be –
first_element = A(1,1)^2 + B(1,1)^2 -2*A(1,1)* B(1,1) + ...
A(1,2)^2 + B(1,2)^2 -2*A(1,2)* B(1,2) … Equation (1)
Now, according to our proposed matrix multiplication, if you check the output of A_ext and B_ext after the loop in the earlier code ends, they would look like the following –
So, if you perform matrix multiplication between A_ext and transpose of B_ext, the first element of the product would be the sum of elementwise multiplication between the first rows of A_ext and B_ext, i.e. sum of these –
The result would be identical to the result obtained from Equation (1) earlier. This would continue for all the elements of A against all the elements of B that are in the same column as in A. Thus, we would end up with the complete squared distance matrix. That’s all there is!!
Vectorized Variations
Vectorized variations of the matrix multiplication based distance matrix calculations are possible, though there weren't any big performance improvements seen with them. Two such variations are listed next.
Variation #1
[nA,dim] = size(A);
nB = size(B,1);
A_ext = ones(nA,dim*3);
A_ext(:,2:3:end) = -2*A;
A_ext(:,3:3:end) = A.^2;
B_ext = ones(nB,dim*3);
B_ext(:,1:3:end) = B.^2;
B_ext(:,2:3:end) = B;
distmat = A_ext * B_ext.';
Variation #2
[nA,dim] = size(A);
nB = size(B,1);
A_ext = [ones(nA*dim,1) -2*A(:) A(:).^2];
B_ext = [B(:).^2 B(:) ones(nB*dim,1)];
A_ext = reshape(permute(reshape(A_ext,nA,dim,[]),[1 3 2]),nA,[]);
B_ext = reshape(permute(reshape(B_ext,nB,dim,[]),[1 3 2]),nB,[]);
distmat = A_ext * B_ext.';
So, these could be considered as experimental versions too.
Use pdist2 (Statistics Toolbox) to compute the distances in a vectorized manner:
ab = im(:,:,2:3); % // get A, B components
ab = reshape(ab, [size(im,1)*size(im,2) 2]); % // reshape into 2-column
dist = pdist2(clusters, ab); % // compute distances
[~, idx] = min(dist); % // find minimizer for each pixel
idx = reshape(idx, size(im,1), size(im,2)); % // reshape result
If you don't have the Statistics Toolbox, you can replace the third line by
dist = squeeze(sum(bsxfun(#minus, clusters, permute(ab, [3 2 1])).^2, 2));
This gives squared distance instead of distance, but for the purposes of minimizing it doesn't matter.
I used the last two dimensions of this 3D matrix as a 2D matrix. So I just wanna multiply the 2D matrix from the result of Matrix1(i,:,:) (which i-by-i) with the vector Matrix2(i,:).') (which is 1-by-i).
The only way I could do that was using an auxiliary matrix that picked up all the numbers from the 2 dimensions from the 3D matrix:
matrixAux(:,:) = Matrix1(1,:,:)
and then I did the multiplication:
matrixAux * (Matrix2(i,:).')
and it worked. However, this is slow because I need to copy all the 3D matrix to a lot of auxiliary matrices, and I need to speed up my code because I'm doing the same operation many times.
How can I do that more efficiently, without having to copy the matrix?
Approach I: bsxfun Multiplication
One approach would be to use the output of bsxfun with #times whose values you can use instead of calculating the matrix multiplication results in a loop -
sum(bsxfun(#times,Matrix1,permute(Matrix2,[1 3 2])),3).'
Example
As an example, let's suppose Matrix1 and Matrix2 are defined like this -
nrows = 3;
p = 6;
ncols = 2;
Matrix1 = rand(nrows,ncols,p)
Matrix2 = rand(nrows,p)
Then, you have your loop like this -
for i = 1:size(Matrix1,1)
matrixAux(:,:) = Matrix1(i,:,:);
matrix_mult1 = matrixAux * (Matrix2(i,:).') %//'
end
So, instead of the loops, you can directly calculate the matrix multiplication results -
matrix_mult2 = sum(bsxfun(#times,Matrix1,permute(Matrix2,[1 3 2])),3).'
Thus, each column of matrix_mult2 would represent matrix_mult1 at each iteration of the loop, as the output of the codes would make it clearer -
matrix_mult1 =
0.7693
0.8690
matrix_mult1 =
1.0649
1.2574
matrix_mult1 =
1.2949
0.6222
matrix_mult2 =
0.7693 1.0649 1.2949
0.8690 1.2574 0.6222
Approach II: "Full" Matrix Multiplication
Now, this must be exciting! Well you can also leverage MATLAB's fast matrix multiplication to get your intermediate matrix multiplication results again without loops. If Matrix1 is nrows x p x ncols, you can reshape it to nrows*p x ncols and then perform matrix multiplication of it with Matrix2. Then, to get an equivalent of matrix_mult2, you need to select indices from the multiplication result. This is precisely achieved here -
%// Get size of Matrix1 to be used regularly inside the codes later on
[m1,n1,p1] = size(Matrix1);
%// Convert 3D Matrix1 to 2D and thus perform "full" matrix multiplication
fmult = reshape(Matrix1,m1*n1,p1)*Matrix2'; %//'
%// Get valid indices
ind = bsxfun(#plus,[1:m1:size(fmult,1)]',[0:nrows-1]*(size(fmult,1)+1)); %//'
%// Get values from the full matrix multiplication result
matrix_mult3 = fmult(ind);
Here, matrix_mult3 must be the same as matrix_mult2.
Observations: Since we are not using all the values calculated from the full matrix multiplication, rather indexing into it and selecting few of its elements, as such this approach performs better than the other approaches under certain circumstances. This approach seems to be the best one when nrows is a small value as we would be using more number of elements from the full matrix multiplication output in that case.
Benchmark Results
Two cases were tested against the three approaches and the test results seem to support our hypotheses discussed earlier.
Case 1
Matrix 1 as 400 x 400 x 400, the runtimes are -
--------------- With Loops
Elapsed time is 2.253536 seconds.
--------------- With BSXFUN
Elapsed time is 0.910104 seconds.
--------------- With Full Matrix Multiplication
Elapsed time is 4.361342 seconds.
Case 2
Matrix 1 as 40 x 2000 x 2000, the runtimes are -
--------------- With Loops
Elapsed time is 5.402487 seconds.
--------------- With BSXFUN
Elapsed time is 2.585860 seconds.
--------------- With Full Matrix Multiplication
Elapsed time is 1.516682 seconds.