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Filtering outlier in 2D coordinate data

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Hi experts,
I need your help to remove some outliers in my 2D coordinate data shown below.
At, at least, four locations, there are some outliers that need to be removed from my data. Please see some spikes in the plots (in red circles).
Does anyone have an algoritm, or can suggest one, that can be used to detect and move (or replace) those values?
Note that I may have several problems with slightly different cases, e.g. the number of points of the outliers may be more or less than the one shown here.
I have attached a txt file containing the 2D coordinate in this post.
Image Analyst
Image Analyst on 17 May 2019
Not really, other than we now know the shapes are called "spikes". We don't know the answers to my questions about whether they are at the same index, or same location.
And also, what if the "spikes" take up a much larger portion of the shape? Like what if the spikes are 30%, 50% or 75% of the image? What parameters do you have restricting the problem. For example if the spikes are taking up 80% then you'd have inward pointing spikes and some algorithms would identify the outward pointing 80% as the spikes. Is that what you want? That the outward pointing ones are always what you want no matter if they end up being the "dominant' smooth shape and not spikes at all?
BeeTiaw on 17 May 2019
Edited: BeeTiaw on 17 May 2019
I can assure you that the spikes will not more than, even, 10%. Let's just stick to the problem I have at the moment. Thanks.

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Accepted Answer

Image Analyst
Image Analyst on 17 May 2019
Well this works for "the problem I have at the moment", though there are even simpler methods, and more sophisticated methods. I think this one is a good mix:
clc; % Clear command window.
clear; % Delete all variables.
close all; % Close all figure windows except those created by imtool.
workspace; % Make sure the workspace panel is showing.
fontSize = 18;
markerSize = 10;
% Read in data.
data = load('Data.txt');
x = data(:,1);
y = data(:,2);
% Smooth the shape with a median filter.
windowWidth = 9;
% for windowWidth = 9 : 2 : 9 % Try different values
smoothedx = medfilt1(x, windowWidth);
smoothedy = medfilt1(y, windowWidth);
plot(x, y, 'b.-', 'MarkerSize', markerSize);
hold on;
plot(smoothedx, smoothedy, 'r.-', 'MarkerSize', markerSize);
grid on;
legend('Original', 'Smoothed');
caption = sprintf('Median Window Width = %d', windowWidth)
title(caption, 'FontSize', fontSize);
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0, 0.04, 1, 0.96]);
% end
% Find where shape deviates from smoothed shape the most.
residuals = sqrt((x-smoothedx) .^ 2 + (y - smoothedy) .^ 2);
plot(residuals, 'b.-', 'MarkerSize', markerSize);
title('Residuals', 'FontSize', fontSize);
grid on;
hold on;
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0, 0.04, 1, 0.96]);
% Find all residuals more than 0.01.
spikeIndexes = residuals > 0.01;
xSpike = x(spikeIndexes);
ySpike = y(spikeIndexes);
% Plot them
plot(x, y, 'b.-');
grid on;
hold on;
plot(xSpike, ySpike, 'ro', 'MarkerSize', markerSize);
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0, 0.04, 1, 0.96]);
0000 Screenshot.png
Image Analyst
Image Analyst on 19 May 2019
You're welcome. Thanks for accepting. Just realize that it may not work for shapes where the spikes are of different widths, and you might need to adjust the parameters until you get exactly 4 regions. You can determine the number of regions with either bwlabel() or findgroups().

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