Why my Gradient calculation shows anomaly?

I have an orginal curve in (X, Y) form in within a limited range (attached).

length(X) = length(Y) = 998

I increased the resolution and extended the range slightly by using PHIP function, as below:

Yintp = Y(1)-0.1:0.002:Y(end)+0.3;
Xintp = pchip(Y, X, Yintp);

Vector size is now, length(X) = length(Y) = 4298

I, then, calculated the gradients for both the orginal (1) and inteprolated (2) curves, as follows;

% Initialize an array to store the gradients
n = length(X);
m = length(Xintp);

gradients1 = zeros(n,1);
gradients2 = zeros(m,1);

% Calculate the forward difference for the first point
gradients1(1) = (X(2) - X(1)) / (Y(2) - Y(1));
gradients2(1) = (Xintp(2) - Xintp(1)) / (Yintp(2) - Yintp(1));

% Calculate the central differences for the middle points
for i = 2:n-1
    gradients1(i) = (X(i+1) - X(i-1)) / (Y(i+1) - Y(i-1));
end

for i = 2:m-1
    gradients2(i) = (Xintp(i+1) - Xintp(i-1)) / (Yintp(i+1) - Yintp(i-1));
end

% Calculate the backward difference for the last point
gradients1(n) = (X(n) - X(n-1)) / (Y(n) - Y(n-1));
gradients2(m) = (Xintp(m) - Xintp(m-1)) / (Yintp(m) - Yintp(m-1));

Displaying results in a figure

% Display the results
figure, 
    plot(Yintp,Xintp); grid on; hold on
    plot(Y, X,'.b','DisplayName','Y vs X low res')
    ylabel('X (cm)')
yyaxis right
    plot(Y, gradients1,'-.r','DisplayName','Gradient - low res');
    plot(Yintp, gradients2,'-', 'DisplayName','Gradient - high res');
    ylabel('Y (cm)')
xlabel('Tangent Angle (deg)')

Problem 1: There is the discontinuity in the 'high res' gradient at the two ends, something not visible in the (X,Y) curves. Below are the expanded views:

Problem 2: Grandient data are noisy and noisiness increases with the resolution, as shown below:

Please help me figure out the dicontinuties in gradient calculations and create smooth gradients from the high res data.

Answer 1

[Epsilon]

Hi Sena,

The ‘smooth’ function can be used before calculating central differences to reduce the noise and improve the stability of the gradient estimates in the given data. To reduce the noise consider smoothening the original data as well.

Exemplar code to smoothen the original and the interpolated data:

X = smooth(X, 0.05, 'rloess');
Y = smooth(Y, 0.05, 'rloess');        
Xintp = smooth(Xintp, 0.05, 'rloess');
Yintp = smooth(Yintp, 0.05, 'rloess');

The plot after smoothening:

Zoomed view with reduced noise:

Link to the documentation on ‘smooth’ function: https://www.mathworks.com/help/curvefit/smooth.html?searchHighlight=smooth&s_tid=srchtitle_support_results_1_smooth

Hope it resolves the issue!

[Sena]

Thank you Epsilon for the great help.

Smoothing factor of 0.05 seems to distort the curve but I will explore the optimal.

[Epsilon]

Yes you can do that, the above is just a typical example. Glad to help.