how can I have a single number as the result of dividing two numbers?

I am implementing this formula in MATLAB:

I[i], is the intensity of my signal and i is the index for each row. My code is as follow:

I attached my signal to this message.

Based on the formula, each time for specific i, I should have a number and then put it in the ith u. but in my code the result of the division is a matrix each time(xc(:,i)./diff((accum))) and I have no idea how to fix it.

M1=length(xc);  
accum1=zeros(M1,1);
accum=zeros(M1+1,1);
a=zeros(1,N);
N=200;
for i=1:N
    for j=i:N
        accum1=accum1+(xc(:,j));
    end
    accum(1:M1) = accum1;
    a(1,i)=xc(:,i)./(2*diff((accum)));
    accum1=zeros(M1,1);
    accum=zeros(M1+1,1);
end

[Chunru]

Can you explain the meaning of Δ in your formula?

It seems that the use of "diff" to accum produces a vector rather than a scalar.

[2NOR_Kh]

this is the definition:

"measurements of I[i] are given by the integrated signal over a small distance, defined by the pixel size Δ and commonly related to the coherence length of the light source"

so I get that "small distance" or saying Δ as diiferentiate function.

But, now that I am thinking, my implementation is not correct. It is saying that delta or Δ is the pixel size, I am not working with pixels but does it mean that this delta is a constant?

Answer 1

[Image Analyst]

Try this:

s = load('matlab.mat')
xc = s.xc;
[rows, columns] = size(xc)
subplot(2, 2, 1);
imshow(xc, []);
subplot(2, 2, 2);
hold on;
grid on;
for col = 1 : columns
    plot(xc(:, col), '-', 'LineWidth', 1);
end
delta = 2; % Whatever it is....
% Compute u for each column.
sumi = cumsum(xc);
for col = 1 : columns
    u = zeros(rows, 1);
    % Reinitialize
    for i = 1 : rows
        % Compute sum from i+1 to the end
        theSum = sumi(end, col) - sumi(i, col);
        u(i) = xc(i) / (2 * delta * theSum);
    end
    subplot(2, 2, 3:4);
    plot(u, '-')
    hold on
end
grid on;

[2NOR_Kh]

the results should be something like (b):

However, maybe somewhere I forgot something.

This formula and figure are from this refrence:

Depth-resolved model-based reconstruction of attenuation coefficients in optical coherence tomography

Answer 2

[Chunru]

load(websave("matalb.mat", "https://www.mathworks.com/matlabcentral/answers/uploaded_files/1104765/matlab.mat"));
[M1, N] = size(xc);  
delta = 1;
% The formula is not well defined for i=M1
% The summation in denominator can be computed using cumsum for efficiency
den = cumsum(flipud(xc(2:end, :)));

a = xc(1:M1-1, :)./(delta*den(end:-1:1, :));

plot(xc(:, 1)); hold on
plot(a(:, 1))