I need to make a scilab / MATLAB program that averages the values of a 3D matrix in cubes of a given size(N x N x N).I am eternally grateful to anyone who can help me.
Thanks in advance
In MATLAB, mat2cell and cellfun make a great team for working on N-dimensional non-overlapping blocks, as I think is the case in the question. An example scenario:
[IN]: A = [30x30x30] array
[IN]: bd = [5 5 5], size of cube
[OUT]: B = [6x6x6] array of block means
To accomplish the above, the solution is:
dims = [30 30 30]; bd = [5 5 5];
A = rand(dims);
f = floor(dims./bd);
remDims = mod(dims,bd); % handle dims that are not a multiple of block size
Ac = mat2cell(A,...
[bd(1)*ones(f(1),1); remDims(1)*ones(remDims(1)>0)], ....
[bd(2)*ones(f(2),1); remDims(2)*ones(remDims(2)>0)], ....
[bd(3)*ones(f(3),1); remDims(3)*ones(remDims(3)>0)] );
B = cellfun(#(x) mean(x(:)),Ac);
If you need a full size output with the mean values replicated, there is a straightforward solution involving the 'UniformOutput' option of cellfun followed by cell2mat.
If you want overlapping cubes and the same size output as input, you can simply do convn(A,ones(blockDims)/prod(blockDims),'same').
EDIT: Simplifications, clarity, generality and fixes.
N = 10; %Same as OP's parameter
M = 10*N;%The input matrix's size in each dimensiona, assumes M is an integer multiple of N
Mat = rand(M,M,M); % A random input matrix
avgs = zeros((M/N)^3,1); %Initializing output vector
l=1; %indexing
for i=1:M/N %indexing 1st coord
for j=1:M/N %indexing 2nd coord
for k=1:M/N % indexing third coord
temp = Mat((i-1)*N+1:i*N,(j-1)*N+1:j*N,(k-1)*N+1:k*N); %temporary copy
avg(l) = mean(temp(:)); %averaging operation on the N*N*N copy
l = l+1; %increment indexing
end
end
end
The for loops and copying can be eliminated once you get the gist of indexing.
Related
I wish to fill a N x M x W matrix ‘S’ with the data from matrices ‘P’ and ‘Q’. They are defined below and illustrated in the attached image. Also, we know for sure that n_1 + n_2 = N, m < M, so all the data may fit in the ‘S’ matrix.
S = zeros(M,N,W);
P = rand(m,n_1,W);
Q = rand(m,n_2,W);
I wish to combine ‘P’ and ‘Q’ in a manner specified by 3 other matrices, ‘Line_num’, ‘P_col’ and ‘Q_col’, described below and in the middle part of the attached image.
P_col = randperm(N); P_col = P_col(1:n_1); % 1 x n_1 matrix
Q_col = setxor(P_col, 1:1:N); % 1 x n_2 matrix
Line_num is a matrix composed of W vectors of the form aa:1:bb, where bb-aa = m and aa is chosen at random for each vector.
The important thing is that in this case the data along the 3rd dimension in all these matrixes represent W different test cases, with the data being different and not to be mixed between each other.
To fill ‘S’ one may proceed in two logical steps (although if it can be done in one I shall be glad)
combine Q and P into an intermediate matrix Y of shape m x N x W by
interweaving their columns. The columns specified in ‘Q_col’ are
taken from Q (using the vector index) and put in the matrix Y (using
the vector value). The same goes for P.
For each of the W vectors composing Line_num and arrays composing S,
use the values in the vector Line_num to spread out Y to the
corresponding rows in S, meanwhile maintaining their top to bottom
order.
I wish to achieve this without for-loops as I am looking to ‘vectorize’ my code and thus improve its running speed.
I have had a look at this post and this post, which are similar to what I desire. However they are simpler as the numbers to be extracted are constant. Maybe something similar would be appropriate?
Thank you for your help :)
Link to the image aforementioned
EDIT: here is an example code with a for-loop of what I want (my problem is that I want to get rid of the loop)
W = 4;
N = 10; n_1 = 4; n_2 = 6;
M = 20; m = 5;
P_col = [1,3,5,8]; % 1 x n_1 matrix
Q_col = setxor(P_col, 1:1:N); % 1 x n_2 matrix
line_num(:,:,1) = [1,5,10,15,18];
line_num(:,:,2) = [2,3,8,11,12];
line_num(:,:,3) = [4,7,8,14,19];
line_num(:,:,4) = [2,6,13,15,16];
S = zeros(M,N,W);
P = rand(m,n_1,W);
Q = rand(m,n_2,W);
for w=1:W
line_num_I = line_num(:,:,w);
S(line_num_I,Q_col,w) = Q(:,:,w);
S(line_num_I,P_col,w) = P(:,:,w);
end
Here is a vectorized solution. I'm not sure if it is more efficient than loop version specially when the size of data is large.
S ( reshape(line_num,[],1,W) ...
+ ([Q_col-1 P_col-1]) * M ...
+ (reshape(0:W-1,1,1,[]))*M*N ...
) ...
= ...
[reshape(Q,[],W);reshape(P,[],W)];
Here implicit expansion is used to convert subscripts to indices. Equivalently bsxfun can be used to compute linear indices:
S ( ...
bsxfun(#plus, ...
reshape(line_num,[],1,W), ...
bsxfun (#plus, ...
([Q_col-1 P_col-1]) * M, ...
(reshape(0:W-1,1,1,[]))*M*N ...
) ...
) ...
) ...
= ...
[reshape(Q,[],W);reshape(P,[],W)];
*Here You can find how to convert 3D index to lindex.
So I ended up finding the answer. For those of you that it may interest, the above for-loop may be replaced by
% 1. Combine columns
mixed_col = zeros(m,N,W);
mixed_col(:,Q_col,:) = Q(:,:,:);
mixed_col(:,P_col,:) = P(:,:,:);
mixed_col = permute(mixed_col,[2,1,3]); % turn 3D matrix into 2D [1]
mixed_col = reshape(mixed_col,N,[],1)';
% 2. Combine lines
S = reshape(S,M*w,N,1); % turn 3D matrix into 2D [2]
line_num_v = permute(line_num + reshape((0:1:(W-1)).*M,1,1,W),[2,1,3]); % turn 3D matrix into 2D [3]
line_num_v = reshape(line_num_v,[],1,1);
S(line_num_v,:) = mixed_col(:,:); % combine using three 2D matrices
S = permute(reshape(S',N,M,W),[2,1,3]);
This involves lots of reshaping but I don't have a simpler answer.
Thanks again for your help.
In MatLab, I have a matrix SimC which has dimension 22 x 4. I re-generate this matrix 10 times using a for loop.
I want to end up with a matrix U that contains SimC(1) in rows 1 to 22, SimC(2) in rows 23 to 45 and so on. Hence U should have dimension 220 x 4 in the end.
Thank you!!
Edit:
nTrials = 10;
n = 22;
U = zeros(nTrials * n , 4) %Dimension of the final output matrix
for i = 1 : nTrials
SimC = SomeSimulation() %This generates an nx4 matrix
U = vertcat(SimC)
end
Unfortunately the above doesn't work as U = vertcat(SimC) only gives back SimC instead of concatenating.
vertcat is a good choice, but it will result in a growing matrix. This is not good practice on larger programs because it can really slow down. In your problem, though, you aren't looping through too many times, so vertcat is fine.
To use vertcat, you would NOT pre-allocate the full final size of the U matrix...just create an empty U. Then, when invoking vertcat, you need to give it both matrices that you want to concatenate:
nTrials = 10;
n = 22;
U = [] %create an empty output matrix
for i = 1 : nTrials
SimC = SomeSimulation(); %This generates an nx4 matrix
U = vertcat(U,SimC); %concatenate the two matrices
end
The better way to do this, since you already know the final size, is to pre-allocate your full U (as you did) and then put your values into U via computing the correct indices. Something like this:
nTrials = 10;
n = 22;
U = U = zeros(nTrials * n , 4); %create a full output matrix
for i = 1 : nTrials
SimC = SomeSimulation(); %This generates an nx4 matrix
indices = (i-1)*n+[1:n]; %here are the rows where you want to put the latest output
U(indices,:)=SimC; %copies SimC into the correct rows of U
end
I have a matrix:
1|2|3|4
4|5|6|7
7|8|9|10
10|11|12|13
I want to multiply the indices of this matrix with indices of another matrix of different size:
7|8|9
9|10|10
10|11|11
for these two matrices I have used the following for loops:
for x=1:4
for y=1:4
for m=1:3
for n=1:3
c=(m*x+n*y);
end
end
end
end
Is there any way to rewrite the above code without using loops? If the indices of each element can be generated in the above matrices, I think it can be done. Please help
mx = m'*x;
mx = mx(:);
ny = n'*y;
ny = ny(:);
mxe = repmat(mx, [length(ny), 1]);
nye = repmat(ny, [length(mx), 1]);
c = mxe+nye;
This will result in c containing all the values that get put in during that loop you have there (note that in your loop, value gets assigned and overwritten).
I would like to concatenate 14641 number of 3X3 matrices into a matrix of size 363X363 and the large matrix must contain 121 submatrices in each row(121*3=363 columns) and 121 such rows of submatrices(121*3=363rows).I have gone through the similar questions but I didn't get the correct logic to concatenate large number of matrices.
Awaiting your suggestions. Thanks in advance.
PS: I got those 3X3 matrices from a 363X363 matrix. The following code is for splitting the single matrix into submatrices.
I=imread('photo.jpg');
B = randi([0 255],363,726,3);
B(1:numel(I)) = I;
L=B(1:363,1:363);
[al,bl]=size(L);
ImageSize=al*bl;
BlockD=3; % i assume 3x3 block
BlockSize=BlockD*BlockD;
TotalBlocks=ImageSize/BlockSize;
subL=zeros(BlockD,BlockD,TotalBlocks); %arrays of blocks.
LL=double(L);
k=1;
for i=1:BlockD:al
for j=1:BlockD:bl
subL(:,:,k)=LL(i:i+BlockD-1,j:j+BlockD-1);
k=k+1;
end
end
Now I want to concatenate all these 'subL' submatrices to form 'LL' again
Using blocproc instead of above code
I tried using blockproc function instead of the above code. I did this piece of code and is working pretty well.Thank you
I=imread('photo.jpg');
B = randi([0 255],363,726,3);
B(1:numel(I)) = I;
L=B(1:363,1:363);
q=[1 2 3 4];
fun=#(block_struct)quaternionrotate(q,block_struct.data);
LL = blockproc(L,[3 3],fun);
and the function quaternionrotate is
function [ Lrot1 ] = quaternionrotate(q,A)
qinv=quatinv(q);
B=zeros(3,1);
A1=[B A];
Lrot=quatmultiply(q,quatmultiply(A1,qinv));
Lrot(:,1)=[];
Lrot1=Lrot;
end
Finally figured this out!
% I used this to make 14641 3x3 sample matrices, you should use your own
matrix = ones(3,3);
for i = 2:14641
matrix = [matrix, i*ones(3,3)];
end
c_size = 121;
output = cell(c_size, c_size) % Make 121x121 cell matrix
% Populate the cell matrix
for i = 1:1:c_size
for j = 1:1:c_size
output(i,j) = {matrix(1:3, (i-1)*c_size*3+(j-1)*3+1 : (i-1)*c_size*3+(j-1)*3+3)};
end
end
This breaks up the 43923x3 size matrix of 363 3x3 matrices into a 121x121 cell matrix of 3x3 matrices. Now that's a tongue twister...
Either way the code does what you need! :)
I have a1 a2 a3. They are constants. I have a matrix A. What I want to do is to get a1*A, a2*A, a3*A three matrices. Then I want transfer them into a diagonal block matrix. For three constants case, this is easy. I can let b1 = a1*A, b2=a2*A, b3=a3*A, then use blkdiag(b1, b2, b3) in matlab.
What if I have n constants, a1 ... an. How could I do this without any looping?I know this can be done by kronecker product but this is very time-consuming and you need do a lot of unnecessary 0 * constant.
Thank you.
Discussion and code
This could be one approach with bsxfun(#plus that facilitates in linear indexing as coded in a function format -
function out = bsxfun_linidx(A,a)
%// Get sizes
[A_nrows,A_ncols] = size(A);
N_a = numel(a);
%// Linear indexing offsets between 2 columns in a block & between 2 blocks
off1 = A_nrows*N_a;
off2 = off1*A_ncols+A_nrows;
%// Get the matrix multiplication results
vals = bsxfun(#times,A,permute(a,[1 3 2])); %// OR vals = A(:)*a_arr;
%// Get linear indices for the first block
block1_idx = bsxfun(#plus,[1:A_nrows]',[0:A_ncols-1]*off1); %//'
%// Initialize output array base on fast pre-allocation inspired by -
%// http://undocumentedmatlab.com/blog/preallocation-performance
out(A_nrows*N_a,A_ncols*N_a) = 0;
%// Get linear indices for all blocks and place vals in out indexed by them
out(bsxfun(#plus,block1_idx(:),(0:N_a-1)*off2)) = vals;
return;
How to use: To use the above listed function code, let's suppose you have the a1, a2, a3, ...., an stored in a vector a, then do something like this out = bsxfun_linidx(A,a) to have the desired output in out.
Benchmarking
This section compares or benchmarks the approach listed in this answer against the other two approaches listed in the other answers for runtime performances.
Other answers were converted to function forms, like so -
function B = bsxfun_blkdiag(A,a)
B = bsxfun(#times, A, reshape(a,1,1,[])); %// step 1: compute products as a 3D array
B = mat2cell(B,size(A,1),size(A,2),ones(1,numel(a))); %// step 2: convert to cell array
B = blkdiag(B{:}); %// step 3: call blkdiag with comma-separated list from cell array
and,
function out = kron_diag(A,a_arr)
out = kron(diag(a_arr),A);
For the comparison, four combinations of sizes of A and a were tested, which are -
A as 500 x 500 and a as 1 x 10
A as 200 x 200 and a as 1 x 50
A as 100 x 100 and a as 1 x 100
A as 50 x 50 and a as 1 x 200
The benchmarking code used is listed next -
%// Datasizes
N_a = [10 50 100 200];
N_A = [500 200 100 50];
timeall = zeros(3,numel(N_a)); %// Array to store runtimes
for iter = 1:numel(N_a)
%// Create random inputs
a = randi(9,1,N_a(iter));
A = rand(N_A(iter),N_A(iter));
%// Time the approaches
func1 = #() kron_diag(A,a);
timeall(1,iter) = timeit(func1); clear func1
func2 = #() bsxfun_blkdiag(A,a);
timeall(2,iter) = timeit(func2); clear func2
func3 = #() bsxfun_linidx(A,a);
timeall(3,iter) = timeit(func3); clear func3
end
%// Plot runtimes against size of A
figure,hold on,grid on
plot(N_A,timeall(1,:),'-ro'),
plot(N_A,timeall(2,:),'-kx'),
plot(N_A,timeall(3,:),'-b+'),
legend('KRON + DIAG','BSXFUN + BLKDIAG','BSXFUN + LINEAR INDEXING'),
xlabel('Datasize (Size of A) ->'),ylabel('Runtimes (sec)'),title('Runtime Plot')
%// Plot runtimes against size of a
figure,hold on,grid on
plot(N_a,timeall(1,:),'-ro'),
plot(N_a,timeall(2,:),'-kx'),
plot(N_a,timeall(3,:),'-b+'),
legend('KRON + DIAG','BSXFUN + BLKDIAG','BSXFUN + LINEAR INDEXING'),
xlabel('Datasize (Size of a) ->'),ylabel('Runtimes (sec)'),title('Runtime Plot')
Runtime plots thus obtained at my end were -
Conclusions: As you can see, either one of the bsxfun based methods could be looked into, depending on what kind of datasizes you are dealing with!
Here's another approach:
Compute the products as a 3D array using bsxfun;
Convert into a cell array with one product (matrix) in each cell;
Call blkdiag with a comma-separated list generated from the cell array.
Let A denote your matrix, and a denote a vector with your constants. Then the desired result B is obtained as
B = bsxfun(#times, A, reshape(a,1,1,[])); %// step 1: compute products as a 3D array
B = mat2cell(B,size(A,1),size(A,2),ones(1,numel(a))); %// step 2: convert to cell array
B = blkdiag(B{:}); %// step 3: call blkdiag with comma-separated list from cell array
Here's a method using kron which seems to be faster and more memory efficient than Divakar's bsxfun based solution. I'm not sure if this is different to your method, but the timing seems pretty good. It might be worth doing some testing between the different methods to work out which is more efficient for you problem.
A=magic(4);
a1=1;
a2=2;
a3=3;
kron(diag([a1 a2 a3]),A)