Fit a polynimial function on a image curve

In image analysis, how can I fit a five or six order polynomial on a waved curve? And how can I determine that polynomial function?

Answer 1

[Image Analyst]

How about this:

% Demo to find fitted curve.
% By Image Analyst
clc;    % Clear the command window.
close all;  % Close all figures (except those of imtool.)
clear;  % Erase all existing variables. Or clearvars if you want.
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 22;

%--------------------------------------------------------------------------------------------------------
%    READ IN IMAGE
folder = pwd;
baseFileName = 'image.jpeg';
% Get the full filename, with path prepended.
fullFileName = fullfile(folder, baseFileName);
% Check if file exists.
if ~exist(fullFileName, 'file')
	% The file doesn't exist -- didn't find it there in that folder.
	% Check the entire search path (other folders) for the file by stripping off the folder.
	fullFileNameOnSearchPath = baseFileName; % No path this time.
	if ~exist(fullFileNameOnSearchPath, 'file')
		% Still didn't find it.  Alert user.
		errorMessage = sprintf('Error: %s does not exist in the search path folders.', fullFileName);
		uiwait(warndlg(errorMessage));
		return;
	end
end
grayImage = imread(fullFileName);

% Get the dimensions of the image.
% numberOfColorChannels should be = 1 for a gray scale image, and 3 for an RGB color image.
[rows, columns, numberOfColorChannels] = size(grayImage);
if numberOfColorChannels > 1
	% It's not really gray scale like we expected - it's color.
	% Use weighted sum of ALL channels to create a gray scale image.
	grayImage = rgb2gray(grayImage);
	% ALTERNATE METHOD: Convert it to gray scale by taking only the green channel,
	% which in a typical snapshot will be the least noisy channel.
	% grayImage = grayImage(:, :, 2); % Take green channel.
end

%--------------------------------------------------------------------------------------------------------
%    SEGMENTATION OF IMAGE
% Get a binary image
binaryImage = imbinarize(grayImage);
subplot(3, 1, 1);
imshow(binaryImage, []);
impixelinfo;
title('Binary Image', 'FontSize', fontSize, 'Interpreter', 'None');
hFig = gcf;
hFig.WindowState = 'maximized'; % May not work in earlier versions of MATLAB.
drawnow;

%--------------------------------------------------------------------------------------------------------
% REMOVE BOTTOM ARC
for col = 1 : columns
	thisCol = binaryImage(:, col);
	% Remove largest blob.
	biggestBlob = bwareafilt(thisCol, 1);
	thisCol(biggestBlob) = false;
	binaryImage(:, col) = thisCol;
end
% Display it.
subplot(3, 1, 2);
imshow(binaryImage)
grid on;
title('Without bottom arc', 'FontSize', fontSize, 'Interpreter', 'None');

% Get (x,y) of what's left
[y, x] = find(binaryImage);
% Add a tiny bit of noise because sometimes there are multiple y for the same x.
x = x + 0.001 * rand(1, length(x));
% Sort them
[x, sortOrder] = sort(x, 'ascend');
y = y(sortOrder);
% Filter with movmean
ySmoothed = movmean(y, 5);
% Display it.
subplot(3, 1, 3);
imshow(binaryImage)
grid on;
title('With Fitted Curve in Red', 'FontSize', fontSize, 'Interpreter', 'None');
hold on
plot(x, ySmoothed, 'r-', 'LineWidth', 3);

Answer 2

[Matt J]

A sixth order polynomial won't give a very good fit, but here is how you could code it:

load waveImage A

[I,J]=find(medfilt2(A,[7,1])-A==-1);
I=size(A,1)+1-I;
p=polyfit(J,I,6);

plot(J,I,'x',J,polyval(p,J),'-')

[mathru]

Thanks for your reply. How can I get the data of wave? Which order polynimal would be good to fit on the wavy surface?

[Image Analyst]

Why do you want a polynomial fit instead of just the smoothed data? You could use something like movmean() or sgolayfilt() to get a curve going through the noisy data. It won't be a perfect polynomial but it will be a better fit to the actual data.

[Matt J]

How can I get the data of wave?

The raw wave data is J,I in my code. In addition to what Image Analyst suggested, you might also consider a cubic smoothing spline,

load waveImage A

[y,x]=find(medfilt2(A,[7,1])-A==-1);
y=size(A,1)+1-y;
 [ yfit,ppfun,M ] = ezsplinefit( x(1:40:end),x,y);

p = 0.001;
sp = csaps(x,y,p);


clf
hold on
fnplt(sp);
plot(x,y,'co');
hold off