I have made an CBIR system in MATLAB and have used similarity measurement as euclidean distance.
Using this for each query image I retrieve top 20 images.
I have used WANG Dataset for testing my system.
It contains 10 classes(like African people, Buses, Roses etc.) each containing 100 images.(1000 images in total).
My Method:
1. I am using Correlogram, Co-occurence Matrix(CCM) and Difference between Pixel Scan Pattern(DBPSP) for constructing my vector(64+196+28=288 dimensions respectively).
Each of the 1000 db image I have its vector constructed beforehand.
Now a query image comes and I construct it's vector too(228 dimensions again).
I use Euclidean Distance for similarity and sort db image vectors in descending order of their euclid distance.
Top 20 results are shown.
In those 20 I can have TP or FP.
For a single query image I can easily calculate Precision and Recall and plot PR-curve using this link.
How can I do the same for whole class?
My Approach: For each image belonging to class A find top 20 images and it's respective TP(true positives) and FP (False Positive).
TP FP
Image1 17 3
Image2 15 5
...
...
Image100 10 10
Total 1500 500
Precision of Class A =1500/(2000) = .75 (Is it right??)
Recall of Class A ---> Stuck ??
PR-Curve ----> Stuck ?? Some links said I need a classifier for that and some not... I am really confused.
So as you noted, you can calculate precision as follows.
P = TP ./ ( TP + FP );
However, you NEED to have either have FN or the number of total falses to calculate recall. As discussed in chat, you need to find a way to determine your FN and FP data. Then you can use the following formula to calculate recall.
R = TP ./ ( TP + FN )
If you have the confusion matrix or data, you can use my custom confusionmat2f1.m to calculate precision, recall, and f1 score. This assumes that the confusion matrix is formatted as how Matlab defines it. An explanation of each line is inline. Please let me know if you want more clarification.
function [F,P,R] = confusionmat2f1( C )
%% confusionmat2f1( C )
%
% Inputs
% C - Confusion Matrix
%
% Outputs
% F - F1 score column vector
% P - Precision column vector
% R - Recall column vector
%%
% Confusion Matrix to Probability
M = sum( C, 3 );
% Calculate Precision
P = diag(M) ./ sum(M,1)';
% Calculate Recall
R = diag(M) ./ sum(M,2);
% Calculate F1 Score
F = f1( P, R );
Related
I have a matrix of 50-by-1 that is demodulated data. As this matrix has only one element in each row, I want to repeat this only element 16 times in each row so the matrix become 50 by 16. I did it using the repmat(A,16) command in Matlab. Now at receiving end noise is also added in matrix of 50 by 16. I want to get it back of 50 by 1 matrix. How can I do this?
I tried averaging of all rows but it is not a valid method. How can I know when an error is occurring in this process?
You are describing a problem of the form y = A * x + n, where y is the observed data, A is a known linear transform, and n is noise. The least squares estimate is the simplest estimate of the unknown vector x. The keys here are to express the repmat() function as a matrix and the observed data as a vector (i.e., a 50*16x1 vector rather than a 50x16 matrix).
x = 10 * rand(50,1); % 50x1 data vector;
A = repmat(eye(length(x)),[16,1]); % This stacks 16 replicas of x.
n = rand(50*16,1); % Noise
y = A * x + n; % Observed data
xhat = A \ y; % Least squares estimate of x.
As for what the inverse (what I assume you mean by 'reverse') of A is, it doesn't have one. If you look at its rank, you'll see it is only 50. The best you can do is to use its pseudoinverse, which is what the \ operator does.
I hope the above helps.
I know that ifft sums multiple sine waves up from data obtain from doing an fft on a signal. is there a way to do a ifft using square waves instead of sine waves?
I'm not trying to get the original signal back but trying to rebuild it using square waves from the data taken from the fft instead of the normal sine wave summation process.
See simple example below: the signals I will be using are human audio signals about 60 seconds long so I'm trying to see if I can use / alter the ifft command in some way.
PS: I'm using Octave 4.0 which is similar to Matlab
clear all,clf reset, clc,tic
Fs = 200; % Sampling frequency
t=linspace(0,1,Fs);
freq=2;
%1 create signal
ya = .5*sin(freq*pi*2*t+pi);
%2 create frequency domain
ya_fft = fft(ya);
%3 rebuild signal
mag = abs(ya_fft);
phase = unwrap(angle(ya_fft));
ya_newifft=ifft(mag.*exp(i*phase));
ifft_sig_combined_L1=ifft(mag.*exp(i*phase),Fs); %use Fs to get correct file length
% square wave
vertoffset=0.5;
A=1
T = 1/freq; % period of the signal
square = mod(t * A / T, A) > A / 2;
square = square - vertoffset;
subplot(3,1,1);
plot(t,ya,'r')
title('orignal signal')
subplot(3,1,2);
plot(t,ifft_sig_combined_L1)
title('rebuilt signal')
subplot(3,1,3);
plot(t,square)
title('rebuilt signal with square wave')
Define the basis vectors you want to use and let them be the columns of a matrix, A. If b is your signal, then just get the least squares solution to Ax = b. If A is full rank, then you will be able to represent b exactly.
Edit:
Think about what a matrix-vector product does: Each column of the matrix is multiplied by the corresponding element of the vector (i.e., the n^th column of the matrix is multiplied by the n^th element of the vector) and the resulting products are summed together. (This would be a lot easier to illustrate if this site supported latex.) In Matlab, a horrible but hopefully illustrative way to do this is
A = some_NxN_matrix;
x = some_Nx1_vector;
b = zeros( size(A,1), 1 );
for n = 1 : length(x)
b = b + A(:,n) * x(n);
end
(Of course, you would never actually do the above but rather b = A*x;.)
Now define whatever square waves you want to use and assign each to its own Nx1 vector. Call these vectors s_1, s_2, ..., s_M, where M is the number of square waves you are using. Now let
A = [s1, s2, ..., s_M];
According to your question, you want to represent your signal as a weighted sum of these square waves. (Note that this is exactly what a DFT does it just uses orthogonal sinusoids rather than square waves.) To weight and sum these square waves, all you have to do is find the matrix-vector product A*x, where x is the vector of coefficients that weight each column (see the above paragraph). Now, if your signal is b and you want to the find the x that will best sum the square waves in order to approximate b, then all you have to do is solve A*x=b. In Matlab, this is given by
x = A \ b;
The rest is just linear algebra. If a left-inverse of A exists (i.e., if A has dimensions M x N and rank N, with M > N), then (A^-1) * A is an identity matrix and
(A^-1) * A * x = (A^-1) * b,
which implies that x = (A^-1) * b, which is what x = A \ b; will return in Matlab. If A has dimensions M x N and rank M, with N > M, then the system is underdetermined and a left-inverse does not exist. In this case you have to use the psuedo-inverse to solve the system. Now suppose that A is NxN with rank N, so that both the left- and right-inverse exist. In this case, x will give an exact representation of b:
x = (A^-1) * b
A * x = A * (A^-1) * b = b
If you want an example of A that uses square waves to get an exact representation of the input signal, check out the Haar transform. There is a function available here.
I am finding it hard to link the theory with the implementation. I would appreciate help in knowing where my understanding is wrong.
Notations - matrix in bold capital and vectors in bold font small letter
is a dataset on observations, each of variables. So, given these observed -dimensional data vectors, the -dimensional principal axes are , for in where is the target dimension.
The principal components of the observed data matrix would be where matrix , matrix , and matrix .
Columns of form an orthogonal basis for the features and the output is the principal component projection that minimizes the squared reconstruction error:
and the optimal reconstruction of is given by .
The data model is
X(i,j) = A(i,:)*S(:,j) + noise
where PCA should be done on X to get the output S. S must be equal to Y.
Problem 1: The reduced data Y is not equal to S that is used in the model. Where is my understanding wrong?
Problem 2: How to reconstruct such that the error is minimum?
Please help. Thank you.
clear all
clc
n1 = 5; %d dimension
n2 = 500; % number of examples
ncomp = 2; % target reduced dimension
%Generating data according to the model
% X(i,j) = A(i,:)*S(:,j) + noise
Ar = orth(randn(n1,ncomp))*diag(ncomp:-1:1);
T = 1:n2;
%generating synthetic data from a dynamical model
S = [ exp(-T/150).*cos( 2*pi*T/50 )
exp(-T/150).*sin( 2*pi*T/50 ) ];
% Normalizing to zero mean and unit variance
S = ( S - repmat( mean(S,2), 1, n2 ) );
S = S ./ repmat( sqrt( mean( Sr.^2, 2 ) ), 1, n2 );
Xr = Ar * S;
Xrnoise = Xr + 0.2 * randn(n1,n2);
h1 = tsplot(S);
X = Xrnoise;
XX = X';
[pc, ~] = eigs(cov(XX), ncomp);
Y = XX*pc;
UPDATE [10 Aug]
Based on the Answer, here is the full code that
clear all
clc
n1 = 5; %d dimension
n2 = 500; % number of examples
ncomp = 2; % target reduced dimension
%Generating data according to the model
% X(i,j) = A(i,:)*S(:,j) + noise
Ar = orth(randn(n1,ncomp))*diag(ncomp:-1:1);
T = 1:n2;
%generating synthetic data from a dynamical model
S = [ exp(-T/150).*cos( 2*pi*T/50 )
exp(-T/150).*sin( 2*pi*T/50 ) ];
% Normalizing to zero mean and unit variance
S = ( S - repmat( mean(S,2), 1, n2 ) );
S = S ./ repmat( sqrt( mean( S.^2, 2 ) ), 1, n2 );
Xr = Ar * S;
Xrnoise = Xr + 0.2 * randn(n1,n2);
X = Xrnoise;
XX = X';
[pc, ~] = eigs(cov(XX), ncomp);
Y = XX*pc; %Y are the principal components of X'
%what you call pc is misleading, these are not the principal components
%These Y columns are orthogonal, and should span the same space
%as S approximatively indeed (not exactly, since you introduced noise).
%If you want to reconstruct
%the original data can be retrieved by projecting
%the principal components back on the original space like this:
Xrnoise_reconstructed = Y*pc';
%Then, you still need to project it through
%to the S space, if you want to reconstruct S
S_reconstruct = Ar'*Xrnoise_reconstructed';
plot(1:length(S_reconstruct),S_reconstruct,'r')
hold on
plot(1:length(S),S)
The plot is which is very different from the one that is shown in the Answer. Only one component of S exactly matches with that of S_reconstructed. Shouldn't the entire original 2 dimensional space of the source input S be reconstructed?
Even if I cut off the noise, then also onely one component of S is exactly reconstructed.
I see nobody answered your question, so here goes:
What you computed in Y are the principal components of X' (what you call pc is misleading, these are not the principal components). These Y columns are orthogonal, and should span the same space as S approximatively indeed (not exactly, since you introduced noise).
If you want to reconstruct Xrnoise, you must look at the theory (e.g. here) and apply it correctly: the original data can be retrieved by projecting the principal components back on the original space like this:
Xrnoise_reconstructed = Y*pc'
Then, you still need to transform it through pinv(Ar)*Xrnoise_reconstructed, if you want to reconstruct S.
Matches nicely for me:
answer to UPDATE [10 Aug]: (EDITED 12 Aug)
Your Ar matrix does not define an orthonormal basis, and as such, the transpose Ar' is not the reverse transformation. The earlier answer I provided was thus wrong. The answer has been corrected above.
Your understanding is quite right. One of the reasons for somebody to use PCA would be to reduce the dimensionality of the data. The first principal component has the largest sample variance amongst of all the normalized linear combinations of the columns of X. The second principal component has maximum variance subject to being orthogonal to the next one, etc.
One might then do a PCA on a dataset, and decide to cut off the last principal component or several of the last principal components of the data. This is done to reduce the effect of the curse of dimensionality. The curse of dimensionality is a term used to point out the fact that any group of vectors is sparse in a relatively high dimensional space. Conversely, this means that you would need an absurd amount of data to form any model on a fairly high dimension dataset, such as an word histogram of a text document with possibly tens of thousands of dimensions.
In effect a dimensionality reduction by PCA removes components that are strongly correlated. For example let's take a look at a picture:
As you can see, most of the values are almost the same, strongly correlated. You could meld some of these correlated pixels by removing the last principal components. This would reduce the dimensionality of the image, pack it, by removing some of the information in the image.
There is no magic way to determine the best amount of principal components or the best reconstruction that I'm aware of.
Forgive me if i am not mathematically rigorous.
If we look at the equation: X = A*S we can say that we are taking some two dimensional data and we map it to a 2 dimensional subspace in 5 dimensional space. Were A is some base for that 2 dimensional subspace.
When we solve the PCA problem for X and look at PC (principal compononet) we see that the two big eignvectors (which coresponds to the two largest eignvalues) span the same subspace that A did. (multpily A'*PC and see that for the first three small eignvectors we get 0 which means that the vectors are orthogonal to A and only for the two largest we get values that are different than 0).
So what i think that the reason why we get a different base for this two dimensional space is because X=A*S can be product of some A1 and S1 and also for some other A2 and S2 and we will still get X=A1*S1=A2*S2. What PCA gives us is a particular base that maximize the variance in each dimension.
So how to solve the problem you have? I can see that you chose as the testing data some exponential times sin and cos so i think you are dealing with a specific kind of data. I am not an expert in signal processing but look at MUSIC algorithm.
You could use the pca function from Statistics toolbox.
coeff = pca(X)
From documentation, each column of coeff contains coefficients for one principal component. So you can reconstruct the observed data X by multiplying with coeff, e.g. X*coeff
I'm trying to reconstruct a 3d image from two calibrated cameras. One of the steps involved is to calculate the 3x3 essential matrix E, from two sets of corresponding (homogeneous) points (more than the 8 required) P_a_orig and P_b_orig and the two camera's 3x3 internal calibration matrices K_a and K_b.
We start off by normalizing our points with
P_a = inv(K_a) * p_a_orig
and
P_b = inv(K_b) * p_b_orig
We also know the constraint
P_b' * E * P_a = 0
I'm following it this far, but how do you actually solve that last problem, e.g. finding the nine values of the E matrix? I've read several different lecture notes on this subject, but they all leave out that crucial last step. Likely because it is supposedly trivial math, but I can't remember when I last did this and I haven't been able to find a solution yet.
This equation is actually pretty common in geometry algorithms, essentially, you are trying to calculate the matrix X from the equation AXB=0. To solve this, you vectorise the equation, which means,
vec() means vectorised form of a matrix, i.e., simply stack the coloumns of the matrix one over the another to produce a single coloumn vector. If you don't know the meaning of the scary looking symbol, its called Kronecker product and you can read it from here, its easy, trust me :-)
Now, say I call the matrix obtained by Kronecker product of B^T and A as C.
Then, vec(X) is the null vector of the matrix C and the way to obtain that is by doing the SVD decomposition of C^TC (C transpose multiplied by C) and take the the last coloumn of the matrix V. This last coloumn is nothing but your vec(X). Reshape X to 3 by 3 matrix. This is you Essential matrix.
In case you find this maths too daunting to code, simply use the following code by Y.Ma et.al:
% p are homogenius coordinates of the first image of size 3 by n
% q are homogenius coordinates of the second image of size 3 by n
function [E] = essentialDiscrete(p,q)
n = size(p);
NPOINTS = n(2);
% set up matrix A such that A*[v1,v2,v3,s1,s2,s3,s4,s5,s6]' = 0
A = zeros(NPOINTS, 9);
if NPOINTS < 9
error('Too few mesurements')
return;
end
for i = 1:NPOINTS
A(i,:) = kron(p(:,i),q(:,i))';
end
r = rank(A);
if r < 8
warning('Measurement matrix rank defficient')
T0 = 0; R = [];
end;
[U,S,V] = svd(A);
% pick the eigenvector corresponding to the smallest eigenvalue
e = V(:,9);
e = (round(1.0e+10*e))*(1.0e-10);
% essential matrix
E = reshape(e, 3, 3);
You can do several things:
The Essential matrix can be estimated using the 8-point algorithm, which you can implement yourself.
You can use the estimateFundamentalMatrix function from the Computer Vision System Toolbox, and then get the Essential matrix from the Fundamental matrix.
Alternatively, you can calibrate your stereo camera system using the estimateCameraParameters function in the Computer Vision System Toolbox, which will compute the Essential matrix for you.
I am new in matlab and I am making a gremetric simulation with k = m2 and p = 1/5.
I have to generate 1000 random numbers and I must show them in a histogram with 15 number of cells. this is what I have so far:
K = 2;
P 1/5;
R = geornd(p,k,1000);
now I am trying to show these result in a histogram with 15 cells but I dont know how to do it please help.
EDIT:
to get the histogram I used:
hist(Sc,15), and this is the results:
According to the doc for geornd, you need to provide the function with a probability parameter P (here 1/5) and a vector dictating the size of the output you want, so it looks like your K is not used correctly in this context.
If you want 1000 random values distributed according to geornd, you might want to use this instead:
R = geornd(0.2,[1 1000]); % P of 0.2 and array of 1 x 1000 numbers
hist(R,15)
Which gives the following:
If you do want do generate 2 distributions, then you can calculate them all at once and plot them separately:
R = geornd(0.2,[2 1000]);
% Plot 1st distribution:
hist(R(1,:),15)
Plot 2nd distribution:
hist(R(2,:),15)