Functions on the Circle

Functions on the circle are \(2\pi\)-periodic functions. Geometrically, one can also denote them as functions on the 1-dimensional torus, or the 1-dimensional sphere.

In MTEX, such functions arise whenever a scalar quantity depends on an in-plane angle. Typical examples are

Representations of Functions on the Circle

In MTEX, functions on the circle can be represented in different ways. If a function is given by an explicit formula, it can be represented by an S1FunHandle. In this representation, the function values are computed directly from the given formula.

Alternatively, a function can be represented by a finite Fourier series using S1FunHarmonic. This representation is discussed in detail in the next section.

As an example, consider the function

\[f(\rho)=1+\frac{1}{2}\cos(2\rho).\]

We define it by an anonymous MATLAB function and pass it to S1FunHandle.

S1F = S1FunHandle(@(rho) 1 + 0.5*cos(2*rho))

S1F = S1FunHandle
  eval: @(rho)1+0.5*cos(2*rho)

The resulting function can be evaluated and plotted like any other function on the circle.

clf
subplot(1,2,1)
plot(S1F,'linewidth',2)
mtexTitle('Polar Plot')
subplot(1,2,2)
plot(S1F,'linewidth',2,'notPolar')
mtexTitle('Cartesian Plot')

Harmonic Representation of Functions on the Circle

Functions on the circle are \(2\pi\)-periodic functions. Hence they may be represented as weighted sums of sines and cosines (Fourier series). In MTEX, we use the numerically more convenient representation in terms of the complex exponentials \(\mathrm e^{-\mathrm i k\rho}\). Hence, a function \(f\) on the circle can be written as series of the form

\[ f(\rho) = \sum_{k=-N}^N \hat f_k \, \mathrm e^{-\mathrm ik\rho} \]

where \(N\) is the bandwidth and \(\hat f_k\) are the Fourier coefficients. Here \(\rho\in[0,2\pi)\) is the angle. MTEX uses the coefficient convention

\[\hat f_k = \frac{1}{2\pi} \int_0^{2\pi} f(\rho)\,\mathrm e^{\mathrm i k\rho}\,\mathrm d\rho.\]

Hence, the constant Fourier coefficient \(\hat f_0\) describes the mean value of \(f\).

Within the class S1FunHarmonic, functions on the circle are represented by their Fourier coefficients, which are stored in the field fun.fhat. As an example, the following Fourier coefficients describe the real-valued function \(f(\rho)=1+\cos\rho+2\sin(2\rho)\).

fhat = [-1i; 0.5; 4; 0.5; 1i];
S1FH = S1FunHarmonic(fhat)

clf
subplot(1,2,1)
plot(S1FH,'linewidth',2)
mtexTitle('Polar Plot')
subplot(1,2,2)
plot(S1FH,'linewidth',2,'notPolar')
mtexTitle('Cartesian Plot')

S1FH = S1FunHarmonic
  bandwidth: 2

Fourier Expansion of an S1FunHandle

An S1FunHandle can be converted into an S1FunHarmonic by computing the corresponding Fourier coefficients. In MTEX, this conversion is performed directly using the command S1FunHarmonic. The desired bandwidth can be specified to balance computational cost and approximation accuracy: a higher bandwidth generally yields a more accurate approximation but requires more computation. Note that, in practice, computations with a bandwidth of 1024 are still very fast and usually provide a good approximation.

S1F = S1FunHandle(@(rho) exp(cos(rho)));
S1FH = S1FunHarmonic(S1F,'bandwidth',20)

clf
plot(S1F,'linewidth',2)
hold on
plot(S1FH,'--','linewidth',2)
hold off
legend('S1FunHandle','S1FunHarmonic')

S1FH = S1FunHarmonic
  bandwidth: 20
  even: true

Interpolation of Sampled Data

Frequently, a function on the circle is known only through its values at a finite set of angles. From these discrete data, MTEX can construct an S1FunHarmonic using the command S1FunHarmonic.interpolate. The resulting trigonometric polynomial provides a periodic interpolation of the sampled values and can subsequently be evaluated at arbitrary angles, plotted, differentiated, or used in further computations.

The bandwidth determines the number of Fourier modes used for the interpolation. It should be chosen sufficiently large to resolve the angular variation of the data, while unnecessarily large bandwidths may introduce oscillations or amplify noise.

rho = linspace(0,2*pi,11).';
values = 1 + cos(rho) + 2*sin(2*rho);

S1FI = S1FunHarmonic.interpolate(rho,values)

clf
plot(rho,values,'x')
hold on
plot(S1FI,'linewidth',2,'noPolar')
hold off

S1FI = S1FunHarmonic
  bandwidth: 3

Smoothing by S1Kernel's

Harmonic functions obtained from measured or interpolated data may contain small-scale oscillations or noise. Such variations can be reduced by convolving the function with an S1DeLaValleePoussinKernel.

The halfwidth of the kernel determines the angular scale of the smoothing. A small halfwidth preserves more local detail, whereas a larger halfwidth produces a smoother function.

In MTEX, smoothing is performed using the command smooth.

f = S1FunHandle(@(rho) 1 + cos(rho) + 0.4*cos(2*rho)+ 0.15*sin(20*rho) + 0.1*cos(35*rho));
f = S1FunHarmonic(f,'bandwidth',64);

fSmooth = f.smooth('halfwidth',8*degree);

clf
plot(f,'linewidth',2,'displayName','original')
hold on
plot(fSmooth,'linewidth',2,'displayName','smoothed')
hold off
legend show

Basic Operations

Standard MATLAB operations can be applied directly to an S1FunHarmonic. The command mean computes the mean value

\[\operatorname{mean}(f) = \frac{1}{2\pi}\int_0^{2\pi} f(\rho)\,\mathrm d\rho,\]

whereas sum returns the integral over the circle,

\[\operatorname{sum}(f) = \int_0^{2\pi} f(\rho)\,\mathrm d\rho = 2\pi\,\operatorname{mean}(f).\]

meanValue = mean(S1FH)
integralValue = sum(S1FH)

meanValue =
    1.2661
integralValue =
    7.9549

The maximum and minimum values and their angular positions can be computed directly.

[maxValue,maxPosition] = max(S1FH)
[minValue,minPosition] = min(S1FH)

maxValue =
    2.7183
maxPosition =
     0
minValue =
    0.3679
minPosition =
    3.1416

Density Estimation from Angular Data

More practically, periodic functions appear after density estimation from circular data, e.g. of the azimuth angle of three dimensional vectors

% some random directions
v = vector3d.rand(1000);

% perform density estimation of the azimuth angle
fun = calcDensity(v.rho,'periodic')

clf
plot(fun,'linewidth',2)

fun = S1FunHarmonic
  bandwidth: 61