Change the size of matrix? - image

I have an image that I'm converting to a binary matrix with size (n,m)
I need a MATLAB function to reshape the size of this matrix to be (n,n).
Otherwise, would it be possible to make the size of image be (n,n) versus the initial (n,m)?

It's actually quite easy. Supposing that your matrix is A and is n x m. I'm assuming you'll want to zero-pad the matrix, meaning that the extra elements would be set to 0. You would simply do this:
[n,m] = size(A);
A(:,m+1:n) = 0;
The first line of code finds the rows n and columns m of the matrix A. Next, we will make all of the rows from the (m+1)th column to the nth column all 0 which effectively makes this a n x n matrix.
Example Run
Here's an example with a 4 x 2 matrix A, and the process requires that we change the size so that A is 4 x 4.
>> A = rand(4,2)
A =
0.9575 0.9572
0.9649 0.4854
0.1576 0.8003
0.9706 0.1419
>> [n,m] = size(A);
>> A(:,m+1:n) = 0
A =
0.9575 0.9572 0 0
0.9649 0.4854 0 0
0.1576 0.8003 0 0
0.9706 0.1419 0 0
Minor Note
This assumes that the number of rows is greater than the number of columns... I'm assuming that this is a requirement on your end. If this is not the case, then the above code won't work. You can make the algorithm agnostic whereas you would zero-pad the matrix in the dimension that has the least amount of entries, but I'll leave that to you as an exercise.

Related

Matrix element replacement Octave

I've a matrix
A = [0.125680 0.543107 0.40088]
If a number in that matrix is greater or equal .5 I want to replace it with 1.
If a number is less than .5 I want to replace it with 0.
So my final matrix will be B = [0 1 0]. How will I do that without for loop?
Get it. It simple.
B = A >= .5;

Psuedo-Random Variable

I have a variable, between 0 and 1, which should dictate the likelyhood that a second variable, a random number between 0 and 1, is greater than 0.5. In other words, if I were to generate the second variable 1000 times, the average should be approximately equal to the first variable's value. How do I make this code?
Oh, and the second variable should always be capable of producing either 0 or 1 in any condition, just more or less likely depending on the value of the first variable. Here is a link to a graph which models approximately how I would like the program to behave. Each equation represents a separate value for the first variable.
You have a variable p and you are looking for a mapping function f(x) that maps random rolls between x in [0, 1] to the same interval [0, 1] such that the expected value, i.e. the average of all rolls, is p.
You have chosen the function prototype
f(x) = pow(x, c)
where c must be chosen appropriately. If x is uniformly distributed in [0, 1], the average value is:
int(f(x) dx, [0, 1]) == p
With the integral:
int(pow(x, c) dx) == pow(x, c + 1) / (c + 1) + K
one gets:
c = 1/p - 1
A different approach is to make p the median value of the distribution, such that half of the rolls fall below p, the other half above p. This yields a different distribution. (I am aware that you didn't ask for that.) Now, we have to satisfy the condition:
f(0.5) == pow(0.5, c) == p
which yields:
c = log(p) / log(0.5)
With the current function prototype, you cannot satisfy both requirements. Your function is also asymmetric (f(x, p) != f(1-x, 1-p)).
Python functions below:
def medianrand(p):
"""Random number between 0 and 1 whose median is p"""
c = math.log(p) / math.log(0.5)
return math.pow(random.random(), c)
def averagerand(p):
"""Random number between 0 and 1 whose expected value is p"""
c = 1/p - 1
return math.pow(random.random(), c)
You can do this by using a dummy. First set the first variable to a value between 0 and 1. Then create a random number in the dummy between 0 and 1. If this dummy is bigger than the first variable, you generate a random number between 0 and 0.5, and otherwise you generate a number between 0.5 and 1.
In pseudocode:
real a = 0.7
real total = 0.0
for i between 0 and 1000 begin
real dummy = rand(0,1)
real b
if dummy > a then
b = rand(0,0.5)
else
b = rand(0.5,1)
end if
total = total + b
end for
real avg = total / 1000
Please note that this algorithm will generate average values between 0.25 and 0.75. For a = 1 it will only generate random values between 0.5 and 1, which should average to 0.75. For a=0 it will generate only random numbers between 0 and 0.5, which should average to 0.25.
I've made a sort of pseudo-solution to this problem, which I think is acceptable.
Here is the algorithm I made;
a = 0.2 # variable one
b = 0 # variable two
b = random.random()
b = b^(1/(2^(4*a-1)))
It doesn't actually produce the average results that I wanted, but it's close enough for my purposes.
Edit: Here's a graph I made that consists of a large amount of datapoints I generated with a python script using this algorithm;
import random
mod = 6
div = 100
for z in xrange(div):
s = 0
for i in xrange (100000):
a = (z+1)/float(div) # variable one
b = random.random() # variable two
c = b**(1/(2**((mod*a*2)-mod)))
s += c
print str((z+1)/float(div)) + "\t" + str(round(s/100000.0, 3))
Each point in the table is the result of 100000 randomly generated points from the algorithm; their x positions being the a value given, and their y positions being their average. Ideally they would fit to a straight line of y = x, but as you can see they fit closer to an arctan equation. I'm trying to mess around with the algorithm so that the averages fit the line, but I haven't had much luck as of yet.

Calculate rank of binary matrix with larger size

Hell all, I have some problem when compute the rank of binary matrix that only 1 or 0. The rank of binary matrix will based on the row reduction using boolean operations XOR. Let see the XOR operation:
1 xor 1 =0
1 xor 0= 1
0 xor 0= 0
0 xor 1= 1
Given a binary matrix as
A =
1 1 0 0 0 0
1 0 0 0 0 1
0 1 0 0 0 1
We can see the third row equals first row xor with second row. Hence, the rank of matrix A only 2, instead of 3 by rank matlab function.
I have one way to compute the extractly rank of binary matrix using this code
B=gf(A)
rank(B)
It will return 2. However, when I compute with large size of matrix, for example 400 by 400. It does not return the rank (never stop). Could you suggest to me the good way to find rank of binary matrix for large size? Thank all
UPDATE: this is computation time using tic toc
N=50; Elapsed time is=0.646823 seconds
N=100;Elapsed time is 3.123573 seconds.
N=150;Elapsed time is 7.438541 seconds.
N=200;Elapsed time is 11.349964 seconds.
N=400;Elapsed time is 66.815286 seconds.
Note that check rank is only the condition in my algorithm. However, it take very long long time, then it will affect to my method
Base on the suggestion of R. I will use Gaussian Elimination to find the rank. This is my code. However, it call the rank function (spend some computation times). Could you modify help me without using rank function?
function rankA=GaussEliRank(A)
mat = A;
[m n] = size(A); % read the size of the original matrix A
for i = 1 : n
j = find(mat(i:m, i), 1); % finds the FIRST 1 in i-th column starting at i
if isempty(j)
mat = mat( sum(mat,2)>0 ,:);
rankA=rank(mat); %%Here
return;
else
j = j + i - 1; % we need to add i-1 since j starts at i
temp = mat(j, :); % swap rows
mat(j, :) = mat(i, :);
mat(i, :) = temp;
% add i-th row to all rows that contain 1 in i-th column
% starting at j+1 - remember up to j are zeros
for k = find(mat( (j+1):m, i ))'
mat(j + k, :) = bitxor(mat(j + k, :), mat(i, :));
end
end
end
%remove all-zero rows if there are some
mat = mat( sum(mat,2)>0 ,:);
if any(sum( mat(:,1:n) ,2)==0) % no solution because matrix A contains
error('No solution.'); % all-zero row, but with nonzero RHS
end
rankA=rank(mat); %%Here
end
Let check the matrix A at here. Correct ans is 393 for rank of A.
Once you get the matrix into row echelon form with Gaussian elimination, the rank is the number of nonzero rows. You should be able to replace the code after the loop with something like rankA=sum(sum(mat,2)>0);.

Enumerate matrix combinations with fixed row and column sums

I'm attempting to find an algorithm (not a matlab command) to enumerate all possible NxM matrices with the constraints of having only positive integers in each cell (or 0) and fixed sums for each row and column (these are the parameters of the algorithm).
Exemple :
Enumerate all 2x3 matrices with row totals 2, 1 and column totals 0, 1, 2:
| 0 0 2 | = 2
| 0 1 0 | = 1
0 1 2
| 0 1 1 | = 2
| 0 0 1 | = 1
0 1 2
This is a rather simple example, but as N and M increase, as well as the sums, there can be a lot of possibilities.
Edit 1
I might have a valid arrangement to start the algorithm:
matrix = new Matrix(N, M) // NxM matrix filled with 0s
FOR i FROM 0 TO matrix.rows().count()
FOR j FROM 0 TO matrix.columns().count()
a = target_row_sum[i] - matrix.rows[i].sum()
b = target_column_sum[j] - matrix.columns[j].sum()
matrix[i, j] = min(a, b)
END FOR
END FOR
target_row_sum[i] being the expected sum on row i.
In the example above it gives the 2nd arrangement.
Edit 2:
(based on j_random_hacker's last statement)
Let M be any matrix verifying the given conditions (row and column sums fixed, positive or null cell values).
Let (a, b, c, d) be 4 cell values in M where (a, b) and (c, d) are on the same row, and (a, c) and (b, d) are on the same column.
Let Xa be the row number of the cell containing a and Ya be its column number.
Example:
| 1 a b |
| 1 2 3 |
| 1 c d |
-> Xa = 0, Ya = 1
-> Xb = 0, Yb = 2
-> Xc = 2, Yc = 1
-> Xd = 2, Yd = 2
Here is an algorithm to get all the combinations verifying the initial conditions and making only a, b, c and d varying:
// A matrix array containing a single element, M
// It will be filled with all possible combinations
matrices = [M]
I = min(a, d)
J = min(b, c)
FOR i FROM 1 TO I
tmp_matrix = M
tmp_matrix[Xa, Ya] = a - i
tmp_matrix[Xb, Yb] = b + i
tmp_matrix[Xc, Yc] = c - i
tmp_matrix[Xd, Yd] = d + i
matrices.add(tmp_matrix)
END FOR
FOR j FROM 1 TO J
tmp_matrix = M
tmp_matrix[Xa, Ya] = a + j
tmp_matrix[Xb, Yb] = b - j
tmp_matrix[Xc, Yc] = c + j
tmp_matrix[Xd, Yd] = d - j
matrices.add(tmp_matrix)
END FOR
It should then be possible to find every possible combination of matrix values:
Apply the algorithm on the first matrix for every possible group of 4 cells ;
Recursively apply the algorithm on each sub-matrix obtained by the previous iteration, for every possible group of 4 cells except any group already used in a parent execution ;
The recursive depth should be (N*(N-1)/2)*(M*(M-1)/2), each execution resulting in ((N*(N-1)/2)*(M*(M-1)/2) - depth)*(I+J+1) sub-matrices. But this creates a LOT of duplicate matrices, so this could probably be optimized.
Are you needing this to calculate Fisher's exact test? Because that requires what you're doing, and based on that page, it seems there will in general be a vast number of solutions, so you probably can't do better than a brute force recursive enumeration if you want every solution. OTOH it seems Monte Carlo approximations are successfully used by some software instead of full-blown enumerations.
I asked a similar question, which might be helpful. Although that question deals with preserving frequencies of letters in each row and column rather than sums, some results can be translated across. E.g. if you find any submatrix (pair of not-necessarily-adjacent rows and pair of not-necessarily-adjacent columns) with numbers
xy
yx
Then you can rearrange these to
yx
xy
without changing any row or column sums. However:
mhum's answer proves that there will in general be valid matrices that cannot be reached by any sequence of such 2x2 swaps. This can be seen by taking his 3x3 matrices and mapping A -> 1, B -> 2, C -> 4 and noticing that, because no element appears more than once in a row or column, frequency preservation in the original matrix is equivalent to sum preservation in the new matrix. However...
someone's answer links to a mathematical proof that it actually will work for matrices whose entries are just 0 or 1.
More generally, if you have any submatrix
ab
cd
where the (not necessarily unique) minimum is d, then you can replace this with any of the d+1 matrices
ef
gh
where h = d-i, g = c+i, f = b+i and e = a-i, for any integer 0 <= i <= d.
For a NXM matrix you have NXM unknowns and N+M equations. Put random numbers to the top-left (N-1)X(M-1) sub-matrix, except for the (N-1, M-1) element. Now, you can find the closed form for the rest of N+M elements trivially.
More details: There are total of T = N*M elements
There are R = (N-1)+(M-1)-1 randomly filled out elements.
Remaining number of unknowns: T-S = N*M - (N-1)*(M-1) +1 = N+M

How to inverse matrix and integer result in Octave?

I would like to get an invertible matrix in Octave but as integers matrix, so:
x = [9,15;19,2];
inv(x)
Here I get:
[-0.0074906, 0.0561798; 0.0711610, -0.0337079]
but I would like to get [22,17;25,21]
anyone knows how to invert a matrix?
The inverse of each element is:
x .^ -1
Which results
0.1111 0.0667
0.0526 0.5000
Why you want to get [22,17;25,21]? What mathematical operation would yield such result?
Invert a matrix in octave:
You are confused about what an inverse of a matrix is, don't nobody here knows what you want with your output, so here are some clues.
If you Invert an identity matrix, you get the identity matrix:
octave:3> a = [1,0;0,1]
a =
1 0
0 1
octave:4> inv(a)
ans =
1 0
0 1
Non-square matrices (m-by-n matrices for which m != n) do not have an inverse
x = [9,15;19,2;5,5];
inv(x)
%error: inverse: argument must be a square matrix
Inverting a matrix with a zero on the diagonal causes an infinity:
octave:5> a = [1,0;0,0]
a =
1 0
0 0
octave:6> inv(a)
warning: inverse: matrix singular to machine precision, rcond = 0
ans =
Inf Inf
Inf Inf
Inverting a matrix with full values like this:
octave:1> a = [1,2;3,4]
a =
1 2
3 4
octave:2> inv(a)
ans =
-2.00000 1.00000
1.50000 -0.50000
For a description of what is happening under the hood of the inverse function:
https://www.khanacademy.org/math/precalculus/precalc-matrices/inverting_matrices/v/inverse-of-a-2x2-matrix
I'm very late to this and don't know how to answer the question efficiently, but it looks like you're looking to find the modular inverse of the matrix, in particular mod 26.
x = [9,15,19,2];
modulus = 26;
inverse_determinant = mod_inverse(det(x),modulus)
You have to implement the mod_inverse function by yourself, but the algorithm should be easy enough to find. If this is only for small modulus values, then a linear search should be efficient enough.
result = mod(det(x)*inv(x)*inverse_determinant,modulus)`

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