I try to implement image compression using Burrows-Wheeler transform. Consider an 1D matrix from path scanning is:
p = [2 5 4 2 3 1 5];
and then apply the Burrows-Wheeler transform :
function output = bwtenc(p)
n = numel(p);
x = zeros(length(p),1);
for i = 1:length(p)
left_cyclic = mod(bsxfun(#plus, 1:n, (0:n-1).')-1, n) + 1;
x = p(left_cyclic);
end
[lex ind] = sortrows(x);
output = lex(:,end);
output = uint8(output(:)');
end
And it works! But the problem is when i try to implement 1D matrix from Lena.bmp which the size is 512*512, Error message showing that bsxfun is out of memory. Anyone please help me.
See if this works for you -
function output = bwtenc(p)
np = numel(p);
[~,sorted_ind] = sort(p);
ind1 = mod((1:np)+np-2,np)+1;
output = p(ind1(sorted_ind));
output = uint8(output(:)');
end
Related
I am trying to write a code for the colculate autocorrelation function. Could you suggest what I'm doing wrong.
Thanks in advance
i = 1:199;
U =sin(0.3*i);
sum_1 = 0;
for tau = 1:length(U)
for t = tau:length(U)-1
sum_1 = sum_1+sum(U(t).*U(t+1-tau));
r(tau)=sum_1;
end
end
You can do something like this. This is an implementation of the original algorithm with only a small optimization; calculating half the sequence and copying the remaining half.
n = length(x);
m = 2*n - 1;
rxx = zeros(1,m);
for i = 1:length(x)
rxx(i) = dot(x(n-i+1:n), x(1:i).');
rxx(m-i+1) = rxx(i)';
end
Example:
x = [2, 3, -1];
rxx =
-2 3 14 3 -2
% MATLAB built-in
xcorr(x,x)
ans =
-2 3 14 3 -2
I have an implementation of a convolution neural network in MATLAB (from the open source DeepLearnToolbox). The following code finds the convolution of different weights and parameters:
z = z + convn(net.layers{l - 1}.a{i}, net.layers{l}.k{i}{j}, 'valid');
To update the tool, I have implemented my own fixed-point scheme based convolution using the following code:
function result = convolution(image, kernal)
% find dimensions of output
row = size(image,1) - size(kernal,1) + 1;
col = size(image,2) - size(kernal,2) + 1;
zdim = size(image,3);
%create output matrix
output = zeros(row, col);
% flip the kernal
kernal_flipped = fliplr(flipud(kernal));
%find rows and col of kernal for loop iteration
row_ker = size(kernal_flipped,1);
col_ker = size(kernal_flipped,2);
for k = 1 : zdim
for i = 0 : row-1
for j = 0 : col-1
sum = fi(0,1,8,7);
prod = fi(0,1,8,7);
for k_row = 1 : row_ker
for k_col = 1 : col_ker
a = image(k_row+i, k_col+j, k);
b = kernal_flipped(k_row,k_col);
prod = a * b;
% convert to fixed point
prod = fi((product/16384), 1, 8, 7);
sum = fi((sum + prod), 1, 8, 7);
end
end
output(i+1, j+1, k) = sum;
end
end
end
result = output;
end
The problem is that when I use my convolution implementation in the bigger application, it is super slow.
Any suggestions how to improve its execution time?
MATLAB doesn't support fixed point 2D convolution, but knowing that convolution can be written as matrix multiplication and that MATLAB has support for fixed point matrix multiplication you can use im2col to convert the image into column format and multiply it by the kernel to convolve them.
row = size(image,1) - size(kernal,1) + 1;
col = size(image,2) - size(kernal,2) + 1;
zdim = size(image,3);
output = zeros(row, col);
kernal_flipped = fliplr(flipud(kernal));
fi_kernel = fi(kernal_flipped(:).', 1, 8, 7) / 16384;
sz = size(kernal_flipped);
sz_img = size(image);
% Use the generated indexes to convert the image into column format
idx_col = im2col(reshape(1:numel(image)/zdim,sz_img(1:2)),sz,'sliding');
image = reshape(image,[],zdim);
for k = 1:zdim
output(:,:,k) = double(fi_kernel * reshape(image(idx_col,k),size(idx_col)));
end
I actually vectorizing one of my code and I have some issues.
This is my initial code:
CoordVorBd = random(N+1,3)
CoordCP = random(N,3)
v = random(1,3)
for i = 1 : N
for j = 1 : N
ri1j = (-CoordVorBd (i,:) + CoordCP(j,:));
vij(i,j,:) = cross(v,ri1j))/(norm(ri1j)
end
end
I have start to vectorize that creating some matrix that contains 3*1 Vectors. My size of matrix is N*N*3.
CoordVorBd1(1:N,:) = CoordVorBd(2:N+1,:);
CoordCP_x= CoordCP(:,1);
CoordCP_y= CoordCP(:,2);
CoordCP_z= CoordCP(:,3);
CoordVorBd_x = CoordVorBd([1:N],1);
CoordVorBd_y = CoordVorBd([1:N],2);
CoordVorBd_z = CoordVorBd([1:N],3);
CoordVorBd1_x = CoordVorBd1(:,1);
CoordVorBd1_y = CoordVorBd1(:,2);
CoordVorBd1_z = CoordVorBd1(:,3);
[X,Y] = meshgrid (1:N);
ri1j_x = (-CoordVorBd_x(X) + CoordCP_x(Y));
ri1j_y = (-CoordVorBd_y(X) + CoordCP_y(Y));
ri1j_z = (-CoordVorBd_z(X) + CoordCP_z(Y));
ri1jmat(:,:,1) = ri1j_x(:,:);
ri1jmat(:,:,2) = ri1j_y(:,:);
ri1jmat(:,:,3) = ri1j_z(:,:);
vmat(:,:,1) = ones(N)*v(1);
vmat(:,:,2) = ones(N)*v(2);
vmat(:,:,3) = ones(N)*v(3);
This code works but is heavy in terms of variable creation. I did'nt achieve to apply the vectorization to all the matrix in one time.
The formule like
ri1jmat(X,Y,1:3) = (-CoordVorBd (X,:) + CoordCP(Y,:));
doesn't work...
If someone have some ideas to have something cleaner.
At this point I have a N*N*3 matrix ri1jmat with all my vectors.
I want to compute the N*N rij1norm matrix that is the norm of the vectors
rij1norm(i,j) = norm(ri1jmat(i,j,1:3))
to be able to vectorize the vij matrix.
vij(:,:,1:3) = (cross(vmat(:,:,1:3),ri1jmat(:,:,1:3))/(ri1jmatnorm(:,:));
The cross product works.
I tried numbers of method without achieve to have this rij1norm matrix without doing a double loop.
If someone have some tricks, thanks by advance.
Here's a vectorized version. Note your original loop didn't include the last column of CoordVorBd, so if that was intentional you need to remove it from the below code as well. I assumed it was a mistake.
CoordVorBd = rand(N+1,3);
CoordCP = rand(N,3);
v = rand(1,3);
repCoordVor=kron(CoordVorBd', ones(1,size(CoordCP,1)))'; %based on http://stackoverflow.com/questions/16266804/matlab-repeat-every-column-sequentially-n-times
repCoordCP=repmat(CoordCP, size(CoordVorBd,1),1); %repeat matrix
V2=-repCoordVor + repCoordCP; %your ri1j
nrm123=sqrt(sum(V2.^2,2)); %vectorized norm for each row
vij_unformatted=cat(3,(v(:,2).*V2(:,3) - V2(:,2).*v(:,3))./nrm123,(v(:,3).*V2(:,1) - V2(:,3).*v(:,1))./nrm123,(v(:,1).*V2(:,2) - V2(:,1).*v(:,2))./nrm123); % cross product, expanded, and each term divided by norm, could use bsxfun(#rdivide,cr123,nrm123) instead, if cr123 is same without divisions
vij=permute(reshape( vij_unformatted,N,N+1,3),[2,1,3]); %reformat to match your vij
Here is another way to do it using arrayfun
% Define a meshgrid of indices to run over
[I, J] = meshgrid(1:N, 1:(N+1));
% Calculate ril for each index
rilj = arrayfun(#(x, y) -CoordVorBd (y,:) + CoordCP(x,:), I, J, 'UniformOutput', false);
%Calculate vij for each point
temp_vij1 = arrayfun(#(x, y) cross(v, rilj{x, y}) / norm(rilj{x, y}), J, I, 'UniformOutput', false);
%Reshape the matrix into desired format
temp_vij2 = cell2mat(temp_vij1);
vij = cat(3, temp_vij2(:, 1:3:end), temp_vij2(:, 2:3:end), temp_vij2(:, 3:3:end));
Given that we have:
x is 2d matrix with size [numSamples x numFeatures]
A is 2d square matrix with size [numFeatures x numFeatures]
B is a 1d vector with size [1 x numFeatures]
I would like to evaluate the following code without a loop: (or in a way faster way)
out = zeros(1,numSamples);
for i = 1:numSamples
res = sum(repmat(B - x(i,:), numSamples, 1)*A.*(x - repmat(x(i,:), numSamples, 1)), 2).^2;
out(i) = var(res);
end
If you have other suggestions on a faster improvement of the above, it is also more than welcome.
You can bsxfun those piece-by-piece for a vectorized solution -
P1 = bsxfun(#minus,B,x)*A;
P2 = bsxfun(#minus,x,permute(x,[3 2 1]));
out = var(squeeze(sum(bsxfun(#times,P1,P2),2)).^2.');
Partially vectorized approach -
P = (bsxfun(#minus,B,x)*A).'; %//'
out = zeros(1,numSamples);
for i = 1:numSamples
out(i) = var((bsxfun(#minus,x,x(i,:))*P(:,i)).^2);
end
I am writing a graphical representation of numerical stability of differential operators and I am having trouble removing a nested for loop. The code loops through all entries in the X,Y, plane and calculates the stability value for each point. This is done by finding the roots of a polynomial of a size dependent on an input variable (length of input vector results in a polynomial 3d matrix of size(m,n,(lenght of input vector)). The main nested for loop is as follows.
for m = 1:length(z2)
for n = 1:length(z1)
pointpoly(1,:) = p(m,n,:);
r = roots(pointpoly);
if isempty(r),r=1e10;end
z(m,n) = max(abs(r));
end
end
The full code of an example numerical method (Trapezoidal Rule) is as follows. Any and all help is appreciated.
alpha = [-1 1];
beta = [.5 .5];
Wind = 2;
Wsize = 500;
if numel(Wind) == 1
Wind(4) = Wind(1);
Wind(3) = -Wind(1);
Wind(2) = Wind(4);
Wind(1) = Wind(3);
end
if numel(Wsize) == 1
Wsize(2) = Wsize;
end
z1 = linspace(Wind(1),Wind(2),Wsize(1));
z2 = linspace(Wind(3),Wind(4),Wsize(2));
[Z1,Z2] = meshgrid(z1,z2);
z = Z1+1i*Z2;
p = zeros(Wsize(2),Wsize(1),length(alpha));
for n = length(alpha):-1:1
p(:,:,(length(alpha)-n+1)) = alpha(n)-z*beta(n);
end
for m = 1:length(z2)
for n = 1:length(z1)
pointpoly(1,:) = p(m,n,:);
r = roots(pointpoly);
if isempty(r),r=1e10;end
z(m,n) = max(abs(r));
end
end
figure()
surf(Z1,Z2,z,'EdgeColor','None');
caxis([0 2])
cmap = jet(255);
cmap((127:129),:) = 0;
colormap(cmap)
view(2);
title(['Alpha Values (',num2str(alpha),') Beta Values (',num2str(beta),')'])
EDIT::
I was able to remove one of the for loops using the reshape command. So;
for m = 1:length(z2)
for n = 1:length(z1)
pointpoly(1,:) = p(m,n,:);
r = roots(pointpoly);
if isempty(r),r=1e10;end
z(m,n) = max(abs(r));
end
end
has now become
gg = reshape(p,[numel(p)/length(alpha) length(alpha)]);
r = zeros(numel(p)/length(alpha),1);
for n = 1:numel(p)/length(alpha)
temp = roots(gg(n,:));
if isempty(temp),temp = 0;end
r(n,1) = max(abs(temp));
end
z = reshape(r,[Wsize(2),Wsize(1)]);
This might be one for loop, but I am still going through the same number of elements. Is there a way to use the roots command on all of my rows at the same time?