How to generate a customized checker board matrix as fast as possible? - matlab

I need a function that creates a checker board matrix with M rows and N columns of P*Q rectangles. I modified the third solution from here to get that:
function [I] = mycheckerboard(M, N, P, Q)
nr = M*P;
nc = N*Q;
i = floor(mod((0:(nc-1))/Q, 2));
j = floor(mod((0:(nr-1))/P, 2))';
r = repmat(i, [nr 1]);
c = repmat(j, [1 nc]);
I = xor(r, c);
it works with no problem:
I=mycheckerboard(2, 3, 4, 3)
I =
0 0 0 1 1 1 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 1 1 1 0 0 0
1 1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1 1
But it's not fast enough since there are lots of calls of this function in a single run. Is there a faster way to get the result? How can I remove floating point divisions and/or calls of the floor function?

Your code is fairly fast for small matrices, but becomes less so as the dimensions get larger. Here's a one-liner using bsxfun and imresize (requires Image Processing toolbox that most have):
m = 2;
n = 3;
p = 4;
q = 3;
I = imresize(bsxfun(#xor, mod(1:m, 2).', mod(1:n, 2)), [p*m q*n], 'nearest')
Or, inspired by #AndrasDeak's use of kron, this is faster with R2015b:
I = kron(bsxfun(#xor, mod(1:m, 2).', mod(1:n, 2)), ones(p, q))
For a small bit more speed, the underlying code for kron can be simplified by taking advantage of the structure of the problem:
A = bsxfun(#xor, mod(1:m, 2).', mod(1:n, 2));
A = permute(A, [3 1 4 2]);
B = ones(q, 1, p);
I = reshape(bsxfun(#times, A, B), [m*n p*q]);
or as one (long) line:
I = reshape(bsxfun(#times, permute(bsxfun(#xor, mod(1:m, 2).', mod(1:n, 2)), [3 1 4 2]), ones(q, 1, p)), [m*n p*q]);

I suggest first creating a binary matrix for the checkerboard's fields, then using the built-in kron to blow it up to the necessary size:
M = 2;
N = 3;
P = 4;
Q = 3;
[iM,iN] = meshgrid(1:M,1:N);
A = zeros(M,N);
A(mod(iM.'+iN.',2)==1) = 1;
board = kron(A,ones(P,Q))

Related

Create matrix with N-4 rows and N-2 columns

Would you give me any comments on this question? I need to create a matrix in Matlab, following the description below:
for a given N and three values a, b and c. I want to create a (N-4)x(N-2) matrix, e.g. for N=8
a b c 0 0 0
0 a b c 0 0
0 0 a b c 0
0 0 0 a b c
Thanks a Lot
What you are trying to create is a toeplitz matrix. One you know the name, it's simple.
%construct the first row
row=zeros(N-2,1);
row(1:3)=[a,b,c];
%construct the first column
col=zeros(N-4,1);
col(1)=a;
%call the function
toeplitz(col,row)
Result with a=4;b=2;c=3;N=8;
ans =
4 2 3 0 0 0
0 4 2 3 0 0
0 0 4 2 3 0
0 0 0 4 2 3
You could solve using the identity matrix function eye() and some matrix concatenation:
N = 8;
a = 1;
b = 2;
c = 3;
a_I = [zeros(N-4, 0), a*eye(N-4), zeros(N-4, 2)];
b_I = [zeros(N-4, 1), b*eye(N-4), zeros(N-4, 1)];
c_I = [zeros(N-4, 2), c*eye(N-4), zeros(N-4, 0)];
X = a_I + b_I + c_I
Which produces the output
X =
1 2 3 0 0 0
0 1 2 3 0 0
0 0 1 2 3 0
0 0 0 1 2 3
You can also make it slightly more general and let the scalars a, b and c be contained in a vector:
N = 8;
v = [1 2 3];
X = zeros(N-4, N-2);
for ii = 1:length(v)
X = X + [zeros(N-4, ii-1), v(ii)*eye(N-4), zeros(N-4, length(v)-ii)];
end
Which produces the same output.
try this fully vectorized function:
function b=bandmat(a, N, M)
% license: CC-BY
% example: bandmat([1,2,3],4,6)
b=a(:).';
b(M+1)=0;
b=repmat(b,1,N);
b=reshape(b(1:end-N),M,N).';

Perform "outer product" of 2-D matrix and return a 3-D array in MATLAB

I would like to do an operation on a 2-D matrix which somehow looks like the outer product of a vector. I already have written some codes for this task, but it is pretty slow, so I would like to know if there is anything I can do to accelerate it.
I would like to show the code I wrote first, followed by an example to illustrate the task I wanted to do.
My code, version row-by-row
function B = outer2D(A)
B = zeros(size(A,1),size(A,2),size(A,2)); %Pre-allocate the output array
for J = 1 : size(A,1)
B(J,:,:) = transpose(A(J,:))*A(J,:); %Perform outer product on each row of A and assign to the J-th layer of B
end
end
Using the matrix A = randn(30000,20) as the input for testing, it spends 0.317 sec.
My code, version page-by-page
function B = outer2D(A)
B = zeros(size(A,1),size(A,2),size(A,2)); %Pre-allocate the output array
for J = 1 : size(A,2)
B(:,:,J) = repmat(A(:,J),1,size(A,2)).*A; %Evaluate B page-by-page
end
end
Using the matrix A = randn(30000,20) as the input for testing, it spends 0.146 sec.
Example 1
A = [3 0; 1 1; 1 0; -1 1; 0 -2]; %A is the input matrix.
B = outer2D(A);
disp(B)
Then I would expect
(:,:,1) =
9 0
1 1
1 0
1 -1
0 0
(:,:,2) =
0 0
1 1
0 0
-1 1
0 4
The first row of B, [9 0; 0 0], is the outer product of [3 0],
i.e. [3; 0]*[3 0] = [9 0; 0 0].
The second row of B, [1 1; 1 1], is the outer product of [1 1],
i.e. [1; 1]*[1 1] = [1 1; 1 1].
The third row of B, [1 0; 0 0], is the outer product of [1 0],
i.e. [1; 0]*[1 0] = [1 0; 0 0].
And the same for the remaining rows.
Example 2
A =
0 -1 -2
0 1 0
-3 0 2
0 0 0
1 0 0
B = outer2D(A)
disp(B)
Then, similar to the example 1, the expected output is
(:,:,1) =
0 0 0
0 0 0
9 0 -6
0 0 0
1 0 0
(:,:,2) =
0 1 2
0 1 0
0 0 0
0 0 0
0 0 0
(:,:,3) =
0 2 4
0 0 0
-6 0 4
0 0 0
0 0 0
Because the real input in my project is like in the size of 30000 × 2000 and this task is to be performed for many times. So the acceleration of this task is quite essential for me.
I am thinking of eliminating the for-loop in the function. May I have some opinions on this problem?
With auto expansion:
function B = outer2D(A)
B=permute(permute(A,[3 1 2]).*A',[2 3 1]);
end
Without auto expansion:
function B = outer2Dold(A)
B=permute(bsxfun(#times,permute(A,[3 1 2]),A'),[2 3 1]);
end
Outer products are not possible in the matlab language.

Create a horizontically stretched upper triangular matrix

I'd like to create a 4x12 matrix which is very similar to a upper triangle matrix, it looks like this:
1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1
So my question is. What is the most efficient way to create it? no loops, no cellfun. Thanks.
One vectorized approach -
nrows = 4;
ncols = 12;
row_idx = repmat(1:nrows,ncols/nrows,1)
out = bsxfun(#le,[1:nrows]',row_idx(:).')
The Matlab R2015a and later approach using the newly introduced repelem:
n = 4;
m = 3;
out = repelem(triu(ones(n)),1,m);
out =
1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1
It even seems faster than the bsxfun approach, though I can't believe this ;)
Benchmark
Unfortunately I couldn't consider andrew's solution as it is not complete and I didn't got it totally.
function [t] = bench()
n = 4;
m = 12;
t = zeros(3,15);
for ii = 1:15
fcns = {
#() thewaywewalk(ii*n,ii*m);
#() Divakar(ii*n,ii*m);
#() LuisMendo(ii*n,ii*m);
};
% timeit
for jj = 1:100;
t(:,ii) = t(:,ii) + cellfun(#timeit, fcns);
end
end
plot(1:15,t(1,:)); hold on;
plot(1:15,t(2,:)); hold on;
plot(1:15,t(3,:)); hold on;
xlabel('Matrix size: n = x*4, m = x*12')
ylabel('timing')
legend({'thewaywewalk','Divakar','Luis Mendo'},'location','northwest')
end
function Z = thewaywewalk(n,m)
Z = repelem(triu(ones(n)),1,m/n);
end
function Z = Divakar(n,m)
row_idx = repmat(1:n,m/n,1);
Z = bsxfun(#le,[1:n]',row_idx(:).');
end
function Z = LuisMendo(n,m)
Z = reshape(repmat(permute(triu(ones(n,n)), [1 3 2]), [1 m/n 1]), [n m]);
end
First bottomline - small matrices:
The new repelem does a very good job, but also the reshape(repmat(permute... does not disappoint. The bsxfun approach stays a little behind for some medium-sized matrices, before it becomes the leader for large matrices:
Second bottomline - big matrices:
As predicted by Divakar, bsxfun is the fastest for larger matrices, actually as expected as bsxfun is always the fastest! Interesting that the other two align perfectly, on could guess they almost work the same internally.
Create an upper triangular matrix of ones, permute second and third dimensions, repeat along second dimension, and reshape into desired shape:
m = 4;
n = 12;
result = reshape(repmat(permute(triu(ones(m,m)), [1 3 2]), [1 n/m 1]), [m n]);
depending on your matlab version
m = 4;
n = 12;
dec2bin(bitshift(num,-1*[0:n/m:n-1])) %this prints out a string
these should be logical arrays (i don't have either of these so I cant test it)
decimalToBinaryVector(bitshift(num,-1*[0:n/m:n-1]))
de2bi(bitshift(num,-1*[0:n/m:n-1]))

Conditionally replace neighbouring cells

Lets say I have a matrix A:
A =
0 1 0 0
0 0 0 0
0 0 0 1
0 0 0 0
And I want to create a new matrix B of the same dimension where all ones and accompanying neighbours are replaced by the following matrix:
X =
1 1 1
1 2 1
1 1 1
The 2 in matrix X should be placed 'on top' of the 1 values as to get:
B =
1 2 1 0
1 1 2 1
0 0 1 2
0 0 1 1
Values should be added up where elements overlap and matrix X should be 'cut off' in places where it extends the dimensions of matrix A/B The idea is to eventually replace X by a 2d gaussian distribution and matrix A will be large containing many more ones. So it's essential that the code is efficient and fast. This is the code i came up with:
A = [0 1 0 0;0 0 0 0;0 0 0 1;0 0 0 0]
X = [1 1 1;1 2 1;1 1 1]
B = zeros(4,4);
t=1;
indA = find(A==1);
indX = find(X==2);
all = find(X>0);
[iall jall] = ind2sub(size(X),all);
[ia ja] = ind2sub(size(A),indA)
[ix jx] = ind2sub(size(X),indX)
iv = ia-ix
jv = ja-jx
for t=1:numel(iv),
ib = iall+iv(t);
jb = jall+jv(t);
ibjb = [ib(:), jb(:)]
c1 = (ibjb(:,1)>4)|(ibjb(:,1)<1); c2 = (ibjb(:,2)>4)|(ibjb(:,1)<1);
ibjb((c1|c2),:)=[]
isel = ibjb(:,1)-iv(t)
jsel = ibjb(:,2)-jv(t)
B(ibjb(:,1), ibjb(:,2)) = B(ibjb(:,1), ibjb(:,2))+ X(isel, jsel)
t=t+1;
end
Is there a more efficient/faster way (minimizing the loops) to code this function?
What you want is a (2D) convolution. So use conv2:
B = conv2(A, X, 'same');

MATLAB Expanding A Matrix with Zeros

I need a matrix of nxn, where the first pxp of it contains ones and rest are zeros. I can do it with traversing the cells, so I'm not asking a way to do it. I'm looking for "the MATLAB way" to do it, using built-in functions and avoiding loops etc.
To be more clear;
let n=4 and p=2,
then the expected result is:
1 1 0 0
1 1 0 0
0 0 0 0
0 0 0 0
There are possibly more than one elegant solution to do it, so I will accept the answer with the shortest and most readable one.
P.S. The question title looks a bit irrelevant: I put that title because my initial approach would be creating a pxp matrix with ones, then expanding it to nxn with zeros.
The answer is creating a matrix of zeroes, and then setting part of it to 1 using indexing:
For example:
n = 4;
p = 2;
x = zeros(n,n);
x(1:p,1:p) = 1;
If you insist on expanding, you can use:
padarray( zeros(p,p)+1 , [n-p n-p], 0, 'post')
Another way to expand the matrix with zeros:
>> p = 2; n = 4;
>> M = ones(p,p)
M =
1 1
1 1
>> M(n,n) = 0
M =
1 1 0 0
1 1 0 0
0 0 0 0
0 0 0 0
You can create the matrix easily by concatenating horizontally and vertically:
n = 4;
p = 2;
MyMatrix = [ ones(p), zeros(p, n-p); zeros(n-p, n) ];
>> p = 2; n = 4;
>> a = [ones(p, 1); zeros(n - p, 1)]
a =
1
1
0
0
>> A = a * a'
A =
1 1 0 0
1 1 0 0
0 0 0 0
0 0 0 0