How to smooth the line curves in the plot?

I'm using the function "smooth" and ''abs'' to smooth the line curve (I_para) but can't acheive the smooth curves in the plot. I used the smooth function previously and got the curve smooth but this time it doesn't work for my case. Please your help willl be appreciated. Thank you

clear, clc, close all
a_ref = 0.05;
Reff_ref = 20;
z_bottom = 1000:1000:4000;
FOV = [0.2, 0.5, 1, 2, 5, 10];
cloud = 'Sub';

for h = 1:length(z_bottom)
    main_fig = figure('NumberTitle', 'off', 'Name', ['Height ', num2str(z_bottom(h)), 'm']);
    Area = []; % for sort
    COLOR = {'r', 'g', 'b', [0.8500 0.3250 0.0980], [0.4660 0.6740 0.1880], [0 0.4470 0.7410]};
    ylim_max = 0;

    for a = 0.01:0.01:a_ref
        for r = 4:Reff_ref
            clf(main_fig); % Clear the current figure
            subplot(1, 1, 1); % Create subplot for all FOVs
            hold on;
            for f = 1:length(FOV)
                Height = genHeight(z_bottom(h))';
                load(['MCdatabase_', cloud, '/', num2str(a), '-', num2str(r), 'um/', num2str(z_bottom(h)), 'm-', num2str(FOV(f)), 'mrad/I0.mat']);
                load(['MCdatabase_', cloud, '/', num2str(a), '-', num2str(r), 'um/', num2str(z_bottom(h)), 'm-', num2str(FOV(f)), 'mrad/Q0.mat']);
                I_para = abs(1/2 * (I0 + Q0));
                I_per = abs(1/2 * (I0 - Q0));
                
                % Plotting all FOV plots on a single figure for each cloud height
                plot(Height, smooth(abs(sum(I_para, 2))) .* Height.^2, 'Color', COLOR{f}, 'LineWidth', 1.5);
                hold on
                plot(Height, smooth(abs(sum(I_per, 2))) .* Height.^2, 'Color', COLOR{f}, 'LineStyle', '--', 'LineWidth', 1.5);
                title(['Sub-adiabatic cloud, Height ', num2str(z_bottom(h)), 'm'])

                % Add annotation for polarized signals
                if f == 1
                    annotation('textbox', [0.15, 0.15, 0.1, 0.1], 'String', 'P_{\perp}', 'Color', 'k', 'EdgeColor', 'none', 'FontWeight', 'bold', 'FontSize', 12);
                    annotation('textbox', [0.24, 0.5, 0.1, 0.1], 'String', 'P_{\mid\mid}', 'Color', 'k', 'LineStyle', '--', 'EdgeColor', 'none', 'FontWeight', 'bold', 'FontSize', 12);
                end
                Area = [Area, [z_bottom(h); FOV(f); pi * ((2.5 * cot(FOV(f) * 1e-3/2) + z_bottom(h)) * tan(FOV(f) * 1e-3/2))^2]];

                % Calculate the maximum y-axis limit
                ylim_max = max(ylim_max, max(sum(I_para, 2) .* Height.^2));
                ylim_max = max(ylim_max, max(sum(I_per, 2) .* Height.^2));
            end
            hold off;
        end
    end

    % Figure settings based on the data
    zoom on
    xlabel('Height of the cloud (m)','FontSize',14,'FontWeight','normal');
    ylabel('Backscattering signal (Unit)','FontSize',14,'FontWeight','normal');
    set(gca,'color','w','Fontsize',12,'LineWidth',1.5,'Fontweight','normal');
    set(gca,'box','off','Fontname','Arial','Fontsmoothing','on');

    % Convert y-axis to logarithmic scale
    % set(gca, 'YScale', 'log');

    % Define the offset for increasing x-axis limit
    if h == 2
        offset = 2; % Offset for h == 2
    elseif h == 3
        offset = 4; % Offset for h == 3
    else
        offset = 0; % No offset for other cases
    end

    % Set x-axis limits and ticks with a difference of 50 for all figures
    xlim_min = z_bottom(h);
    xlim_max = max(Height) + offset; % Adjusted xlim_max with an offset

    % Adjust xticks for last tick visibility
    if h == 1
        xticks = 1000:50:(max(Height));
    elseif h == 2
        xticks = [2000:50:(max(Height) + offset)]; % Corrected x-ticks for Figure 2
    elseif h == 3
        xticks = [3000:50:(max(Height) + offset)]; % Corrected x-ticks for Figure 3
    else
        xticks = 4000:50:xlim_max;
    end

    set(gca,'xlim',[xlim_min, xlim_max],'xtick',xticks,'ylim',[0 ylim_max],'ytick',[0:0.02:ylim_max]);

    set(gca,'xgrid','on','ygrid','on','gridcolor','k');

    % Create legend entries for each FOV value
    legend_labels = cell(1, length(FOV) * 2); % Preallocate space for legend labels
    for i = 1:length(FOV)
        legend_labels{(i-1)*2+1} = [num2str(FOV(i)), 'mrad'];
        legend_labels{(i-1)*2+2} = [num2str(FOV(i)), 'mrad'];
    end

    % Plot the legend entries in two rows
    legend(legend_labels, 'Location', 'NorthEast', 'NumColumns', 3);
    legend boxoff
end

Answer 1

[Image Analyst]

See attached demo. Adapt as needed.

% Demo to show spline interpolation.
% Clean up / initialize
clc; 
close all; 
clear all; 
workspace; % Display workspace panel.
% Create the original knot points.
lengthX = 10;
x = 1:lengthX;
y = rand (lengthX,1);
% Plot it and show how the line has sharp bends.
plot(x, y, '-sr', 'LineWidth', 2);
set(gcf, 'Position', get(0,'Screensize')); % Maximize figure.
% Use splines to interpolate a smoother curve,
% with 10 times as many points,
% that goes exactly through the same data points.
samplingRateIncrease = 10;
newXSamplePoints = linspace(1, max(x), lengthX * samplingRateIncrease);
smoothedY = spline(x, y, newXSamplePoints);
% Plot smoothedY and show how the line is
% smooth, and has no sharp bends.
hold on; % Don't destroy the first curve we plotted.
plot(newXSamplePoints, smoothedY, '-ob');
title('Spline Interpolation Demo', 'FontSize', 20);
legend('Original Points', 'Spline Points');
% Mathworks Demo code from their Help
% x = 0:10;
% y = sin(x);
% xx = 0:.25:10;
% yy = spline(x,y,xx);
% plot(x,y,'o',xx,yy)
slopes = [0, diff(smoothedY)];
plot(newXSamplePoints, slopes, 'k-', 'LineWidth', 3);
% Draw x axis
line(xlim, [0,0], 'Color', 'k', 'LineWidth', 2);
grid on;
legend('Original Points', 'Spline Points', 'Slope');

[Ram Prasanth]

Yes it is exactly as @Image Analyst Demo example.

You have to plot it with the increased new points which you obtained with the interpolation.

or you can follow the demo and use it as an inspiration for your function.

[Wiqas Ahmad]

I increased the samplingRateIncrease = 1000; but still doesn't work. The portion of the code is here

% Smooth the signal
                samplingRateIncrease = 10000;
                newXSamplePoints = linspace(1, size(I_para, 1), size(I_para, 1) * samplingRateIncrease);
                smoothedY = interp1(1:size(I_para, 1), sum(I_para, 2) .* Height.^2, newXSamplePoints,'linear');

                % Interpolate smoothedY to match the length of Height
                x_interp = linspace(1, numel(smoothedY), numel(Height));
                smoothedY_interp = interp1(1:numel(smoothedY), smoothedY, x_interp, 'linear');

                % Plotting
                plot(Height, smoothedY_interp, 'Color', COLOR{f}, 'LineWidth', 1.5)
                hold on
                plot(Height, smooth(sum(I_per, 2)) .* Height.^2, 'Color', COLOR{f}, 'LineStyle', '--', 'LineWidth', 1.5)
                title(['Sub-adiabatic cloud, Height ', num2str(z_bottom(h)), 'm'])

[Ram Prasanth]

Hello @Wiqas Ahmad?

You need to do few Modifications as i suggested before in your code.

I have changed the necessary and you can adopt it in your code.

smoothedY = interp1(1:size(I_para, 1), sum(I_para, 2 ...
    ) .* Height.^2, newXSamplePoints,'spline'); % Use spline interpolation
% Interpolate smoothedY to match the length of Height
smoothedY_interp = interp1(1:numel(smoothedY), smoothedY, linspace( ...
    1, numel(smoothedY), numel(Height)), 'spline');


samplingRateIncrease = 10000; % Try increasing this value for smoother curves only 
% after testing the spline interpolation method.

Answer 2

[Ram Prasanth]

You can use the spline interpolation method in matlab.

Which allow you to interpolate the values inbetween and increases the number of points, Hence giving you a smooth spline curve.

Here is the documentation for it - 1-D data interpolation (table lookup) - MATLAB interp1 - MathWorks France

Also you can look in to the Spline function in matlab which can be useful too - Cubic spline data interpolation - MATLAB spline - MathWorks France

[Wiqas Ahmad]

I checked but it doesn't work for my case.

[Image Analyst]

How many data points does your smoothed array have? Like he said, you must use more points than your original data or else you will still have "kinks" in your curve.