1.5. SciPy : high-level scientific computing¶

Authors: Gaël Varoquaux, Adrien Chauve, Andre Espaze, Emmanuelle Gouillart, Ralf Gommers

Tip

scipy can be compared to other standard scientific-computing libraries, such as the GSL (GNU Scientific Library for C and C++), or Matlab’s toolboxes. scipy is the core package for scientific routines in Python; it is meant to operate efficiently on numpy arrays, so that NumPy and SciPy work hand in hand.

Before implementing a routine, it is worth checking if the desired data processing is not already implemented in SciPy. As non-professional programmers, scientists often tend to re-invent the wheel, which leads to buggy, non-optimal, difficult-to-share and unmaintainable code. By contrast, SciPy’s routines are optimized and tested, and should therefore be used when possible.

Warning

This tutorial is far from an introduction to numerical computing. As enumerating the different submodules and functions in SciPy would be very boring, we concentrate instead on a few examples to give a general idea of how to use scipy for scientific computing.

scipy is composed of task-specific sub-modules:

`scipy.cluster`	Vector quantization / Kmeans
`scipy.constants`	Physical and mathematical constants
`scipy.fft`	Fourier transform
`scipy.integrate`	Integration routines
`scipy.interpolate`	Interpolation
`scipy.io`	Data input and output
`scipy.linalg`	Linear algebra routines
`scipy.ndimage`	n-dimensional image package
`scipy.odr`	Orthogonal distance regression
`scipy.optimize`	Optimization
`scipy.signal`	Signal processing
`scipy.sparse`	Sparse matrices
`scipy.spatial`	Spatial data structures and algorithms
`scipy.special`	Any special mathematical functions
`scipy.stats`	Statistics

Tip

They all depend on numpy, but are mostly independent of each other. The standard way of importing NumPy and these SciPy modules is:

>>> importnumpyasnp
>>> importscipyassp

1.5.1. File input/output: `scipy.io`¶

scipy.io contains functions for loading and saving data in several common formats including Matlab, IDL, Matrix Market, and Harwell-Boeing.

Matlab files: Loading and saving:

>>> importscipyassp
>>> a=np.ones((3,3))
>>> sp.io.savemat('file.mat',{'a':a})# savemat expects a dictionary
>>> data=sp.io.loadmat('file.mat')
>>> data['a']
array([[1.,  1.,  1.],
       [1.,  1.,  1.],
       [1.,  1.,  1.]])

Warning

Python / Matlab mismatch: The Matlab file format does not support 1D arrays.

>>> a=np.ones(3)
>>> a
array([1.,  1.,  1.])
>>> a.shape
(3,)
>>> sp.io.savemat('file.mat',{'a':a})
>>> a2=sp.io.loadmat('file.mat')['a']
>>> a2
array([[1.,  1.,  1.]])
>>> a2.shape
(1, 3)

Notice that the original array was a one-dimensional array, whereas the saved and reloaded array is a two-dimensional array with a single row.

For other formats, see the scipy.io documentation.

1.5.2. Special functions: `scipy.special`¶

“Special” functions are functions commonly used in science and mathematics that are not considered to be “elementary” functions. Examples include

the gamma function, scipy.special.gamma(),

the error function, scipy.special.erf(),

Bessel functions, such as scipy.special.jv() (Bessel function of the first kind), and

elliptic functions, such as scipy.special.ellipj() (Jacobi elliptic functions).

Other special functions are combinations of familiar elementary functions, but they offer better accuracy or robustness than their naive implementations would.

Most of these function are computed elementwise and follow standard NumPy broadcasting rules when the input arrays have different shapes. For example, scipy.special.xlog1py() is mathematically equivalent to $x\log(1 + y)$ .

>>> importscipyassp
>>> x=np.asarray([1,2])
>>> y=np.asarray([[3],[4],[5]])
>>> res=sp.special.xlog1py(x,y)
>>> res.shape
(3, 2)
>>> ref=x*np.log(1+y)
>>> np.allclose(res,ref)
True

However, scipy.special.xlog1py() is numerically favorable for small $y$ , when explicit addition of 1 would lead to loss of precision due to floating point truncation error.

>>> x=2.5
>>> y=1e-18
>>> x*np.log(1+y)
np.float64(0.0)
>>> sp.special.xlog1py(x,y)
np.float64(2.5e-18)

Many special functions also have “logarithmized” variants. For instance, the gamma function $\Gamma(\cdot)$ is related to the factorial function by $n! = \Gamma(n + 1)$ , but it extends the domain from the positive integers to the complex plane.

>>> x=np.arange(10)
>>> np.allclose(sp.special.gamma(x+1),sp.special.factorial(x))
True
>>> sp.special.gamma(5)<sp.special.gamma(5.5)<sp.special.gamma(6)
np.True_

The factorial function grows quickly, and so the gamma function overflows for moderate values of the argument. However, sometimes only the logarithm of the gamma function is needed. In such cases, we can compute the logarithm of the gamma function directly using scipy.special.gammaln().

>>> x=[5,50,500]
>>> np.log(sp.special.gamma(x))
array([  3.17805383, 144.56574395,          inf])
>>> sp.special.gammaln(x)
array([   3.17805383,  144.56574395, 2605.11585036])

Such functions can often be used when the intermediate components of a calculation would overflow or underflow, but the final result would not. For example, suppose we wish to compute the ratio $\Gamma(500)/\Gamma(499)$ .

>>> a=sp.special.gamma(500)
>>> b=sp.special.gamma(499)
>>> a,b
(np.float64(inf), np.float64(inf))

Both the numerator and denominator overflow, so performing $a / b$ will not return the result we seek. However, the magnitude of the result should be moderate, so the use of logarithms comes to mind. Combining the identities $\log(a/b) = \log(a) - \log(b)$ and $\exp(\log(x)) = x$ , we get:

>>> log_a=sp.special.gammaln(500)
>>> log_b=sp.special.gammaln(499)
>>> log_res=log_a-log_b
>>> res=np.exp(log_res)
>>> res
np.float64(499.0000000...)

Similarly, suppose we wish to compute the difference $\log(\Gamma(500) - \Gamma(499))$ . For this, we use scipy.special.logsumexp(), which computes $\log(\exp(x) + \exp(y))$ using a numerical trick that avoids overflow.

>>> res=sp.special.logsumexp([log_a,log_b],
... b=[1,-1])# weights the terms of the sum
>>> res
np.float64(2605.113844343...)

For more information about these and many other special functions, see the documentation of scipy.special.

1.5.3. Linear algebra operations: `scipy.linalg`¶

scipy.linalg provides a Python interface to efficient, compiled implementations of standard linear algebra operations: the BLAS (Basic Linear Algebra Subroutines) and LAPACK (Linear Algebra PACKage) libraries.

For example, the scipy.linalg.det() function computes the determinant of a square matrix:

>>> importscipyassp
>>> arr=np.array([[1,2],
... [3,4]])
>>> sp.linalg.det(arr)
np.float64(-2.0)

Mathematically, the solution of a linear system $Ax = b$ is $x = A^{-1}b$ , but explicit inversion of a matrix is numerically unstable and should be avoided. Instead, use scipy.linalg.solve():

>>> A=np.array([[1,2],
... [2,3]])
>>> b=np.array([14,23])
>>> x=sp.linalg.solve(A,b)
>>> x
array([4., 5.])
>>> np.allclose(A@x,b)
True

Linear systems with special structure can often be solved more efficiently than more general systems. For example, systems with triangular matrices can be solved using scipy.linalg.solve_triangular():

>>> A_upper=np.triu(A)
>>> A_upper
array([[1, 2],
       [0, 3]])
>>> np.allclose(sp.linalg.solve_triangular(A_upper,b,lower=False),
... sp.linalg.solve(A_upper,b))
True

scipy.linalg also features matrix factorizations/decompositions such as the singular value decomposition.

>>> A=np.array([[1,2],
... [2,3]])
>>> U,s,Vh=sp.linalg.svd(A)
>>> s# singular values
array([4.23606798, 0.23606798])

The original matrix can be recovered by matrix multiplication of the factors:

>>> S=np.diag(s)# convert to diagonal matrix before matrix multiplication
>>> A2=U@S@Vh
>>> np.allclose(A2,A)
True
>>> A3=(U*s)@Vh# more efficient: use array math broadcasting rules!
>>> np.allclose(A3,A)
True

Many other decompositions (e.g. LU, Cholesky, QR), solvers for structured linear systems (e.g. triangular, circulant), eigenvalue problem algorithms, matrix functions (e.g. matrix exponential), and routines for special matrix creation (e.g. block diagonal, toeplitz) are available in scipy.linalg.

1.5.4. Interpolation: `scipy.interpolate`¶

scipy.interpolate is used for fitting a function – an “interpolant” – to experimental or computed data. Once fit, the interpolant can be used to approximate the underlying function at intermediate points; it can also be used to compute the integral, derivative, or inverse of the function.

Some kinds of interpolants, known as “smoothing splines”, are designed to generate smooth curves from noisy data. For example, suppose we have the following data:

>>> rng=np.random.default_rng(27446968)
>>> measured_time=np.linspace(0,2*np.pi,20)
>>> function=np.sin(measured_time)
>>> noise=rng.normal(loc=0,scale=0.1,size=20)
>>> measurements=function+noise

scipy.interpolate.make_smoothing_spline() can be used to form a curve similar to the underlying sine function.

>>> smoothing_spline=sp.interpolate.make_smoothing_spline(measured_time,measurements)
>>> interpolation_time=np.linspace(0,2*np.pi,200)
>>> smooth_results=smoothing_spline(interpolation_time)

../../_images/sphx_glr_plot_interpolation_001.png

On the other hand, if the data are not noisy, it may be desirable to pass exactly through each point.

>>> interp_spline=sp.interpolate.make_interp_spline(measured_time,function)
>>> interp_results=interp_spline(interpolation_time)

../../_images/sphx_glr_plot_interpolation_002.png

The derivative and antiderivative methods of the result object can be used for differentiation and integration. For the latter, the constant of integration is assumed to be zero, but we can “wrap” the antiderivative to include a nonzero constant of integration.

>>> d_interp_spline=interp_spline.derivative()
>>> d_interp_results=d_interp_spline(interpolation_time)
>>> i_interp_spline=lambdat:interp_spline.antiderivative()(t)-1
>>> i_interp_results=i_interp_spline(interpolation_time)

../../_images/sphx_glr_plot_interpolation_003.png

For functions that are monotonic on an interval (e.g. $\sin$ from $\pi/2$ to $3\pi/2$ ), we can reverse the arguments of make_interp_spline to interpolate the inverse function. Because the first argument is expected to be monotonically increasing, we also reverse the order of elements in the arrays with numpy.flip().

>>> i=(measured_time>np.pi/2)&(measured_time<3*np.pi/2)
>>> inverse_spline=sp.interpolate.make_interp_spline(np.flip(function[i]),
... np.flip(measured_time[i]))
>>> inverse_spline(0)
array(3.14159265)

See the summary exercise on Maximum wind speed prediction at the Sprogø station for a more advanced spline interpolation example, and read the SciPy interpolation tutorial and the scipy.interpolate documentation for much more information.

1.5.5. Optimization and fit: `scipy.optimize`¶

scipy.optimize provides algorithms for root finding, curve fitting, and more general optimization.

1.5.5.1. Root Finding¶

scipy.optimize.root_scalar() attempts to find a root of a specified scalar-valued function (i.e., an argument at which the function value is zero). Like many scipy.optimize functions, the function needs an initial guess of the solution, which the algorithm will refine until it converges or recognizes failure. We also provide the derivative to improve the rate of convergence.

>>> deff(x):
... return(x-1)*(x-2)
>>> defdf(x):
... return2*x-3
>>> x0=0# guess
>>> res=sp.optimize.root_scalar(f,x0=x0,fprime=df)
>>> res
      converged: True
           flag: converged
 function_calls: 12
     iterations: 6
           root: 1.0
         method: newton

Warning

None of the functions in scipy.optimize that accept a guess are guaranteed to converge for all possible guesses! (For example, try x0=1.5 in the example above, where the derivative of the function is exactly zero.) If this occurs, try a different guess, adjust the options (like providing a bracket as shown below), or consider whether SciPy offers a more appropriate method for the problem.

Note that only one the root at 1.0 is found. By inspection, we can tell that there is a second root at 2.0. We can direct the function toward a particular root by changing the guess or by passing a bracket that contains only the root we seek.

>>> res=sp.optimize.root_scalar(f,bracket=(1.5,10))
>>> res.root
2.0

For multivariate problems, use scipy.optimize.root().

>>> deff(x):
... # intersection of unit circle and line from origin
... return[x[0]**2+x[1]**2-1,
... x[1]-x[0]]
>>> res=sp.optimize.root(f,x0=[0,0])
>>> np.allclose(f(res.x),0,atol=1e-10)
True
>>> np.allclose(res.x,np.sqrt(2)/2)
True

Over-constrained problems can be solved in the least-squares sense using scipy.optimize.root() with method='lm' (Levenberg-Marquardt).

>>> deff(x):
... # intersection of unit circle, line from origin, and parabola
... return[x[0]**2+x[1]**2-1,
... x[1]-x[0],
... x[1]-x[0]**2]
>>> res=sp.optimize.root(f,x0=[1,1],method='lm')
>>> res.success
True
>>> res.x
array([0.76096066, 0.66017736])

See the documentation of scipy.optimize.root_scalar() and scipy.optimize.root() for a variety of other solution algorithms and options.

1.5.5.2. Curve fitting¶

../../_images/sphx_glr_plot_curve_fit_001.png

Suppose we have data that is sinusoidal but noisy:

>>> x=np.linspace(-5,5,num=50)# 50 values between -5 and 5
>>> noise=0.01*np.cos(100*x)
>>> a,b=2.9,1.5
>>> y=a*np.cos(b*x)+noise

We can approximate the underlying amplitude, frequency, and phase from the data by least squares curve fitting. To begin, we write a function that accepts the independent variable as the first argument and all parameters to fit as separate arguments:

>>> deff(x,a,b,c):
... returna*np.sin(b*x+c)

../../_images/sphx_glr_plot_curve_fit_002.png

We then use scipy.optimize.curve_fit() to find $a$ and $b$ :

>>> params,_=sp.optimize.curve_fit(f,x,y,p0=[2,1,3])
>>> params
array([2.900026  , 1.50012043, 1.57079633])
>>> ref=[a,b,np.pi/2]# what we'd expect
>>> np.allclose(params,ref,rtol=1e-3)
True

1.5.5.3. Optimization¶

../../_images/sphx_glr_plot_optimize_example1_001.png

Suppose we wish to minimize the scalar-valued function of a single variable $f(x) = x^2 + 10 \sin(x)$ :

>>> deff(x):
... returnx**2+10*np.sin(x)
>>> x=np.arange(-5,5,0.1)
>>> plt.plot(x,f(x))
[<matplotlib.lines.Line2D object at ...>]
>>> plt.show()

We can see that the function has a local minimizer near $x = 3.8$ and a global minimizer near $x = -1.3$ , but the precise values cannot be determined from the plot.

The most appropriate function for this purpose is scipy.optimize.minimize_scalar(). Since we know the approximate locations of the minima, we will provide bounds that restrict the search to the vicinity of the global minimum.

>>> res=sp.optimize.minimize_scalar(f,bounds=(-2,-1))
>>> res
 message: Solution found.
 success: True
  status: 0
     fun: -7.9458233756...
       x: -1.306440997...
     nit: 8
    nfev: 8
>>> res.fun==f(res.x)
np.True_

If we did not already know the approximate location of the global minimum, we could use one of SciPy’s global minimizers, such as scipy.optimize.differential_evolution(). We are required to pass bounds, but they do not need to be tight.

>>> bounds=[(-5,5)]# list of lower, upper bound for each variable
>>> res=sp.optimize.differential_evolution(f,bounds=bounds)
>>> res
 message: Optimization terminated successfully.
 success: True
     fun: -7.9458233756...
       x: [-1.306e+00]
     nit: 6
    nfev: 111
     jac: [ 9.948e-06]

For multivariate optimization, a good choice for many problems is scipy.optimize.minimize(). Suppose we wish to find the minimum of a quadratic function of two variables, $f(x_0, x_1) = (x_0-1)^2 + (x_1-2)^2$ .

>>> deff(x):
... return(x[0]-1)**2+(x[1]-2)**2

Like scipy.optimize.root(), scipy.optimize.minimize() requires a guess x0. (Note that this is the initial value of both variables rather than the value of the variable we happened to label $x_0$ .)

>>> res=sp.optimize.minimize(f,x0=[0,0])
>>> res
  message: Optimization terminated successfully.
  success: True
   status: 0
      fun: 1.70578...e-16
        x: [ 1.000e+00  2.000e+00]
      nit: 2
      jac: [ 3.219e-09 -8.462e-09]
 hess_inv: [[ 9.000e-01 -2.000e-01]
            [-2.000e-01  6.000e-01]]
     nfev: 9
     njev: 3

This barely scratches the surface of SciPy’s optimization features, which include mixed integer linear programming, constrained nonlinear programming, and the solution of assignment problems. For much more information, see the documentation of scipy.optimize and the advanced chapter Mathematical optimization: finding minima of functions.

See the summary exercise on Non linear least squares curve fitting: application to point extraction in topographical lidar data for another, more advanced example.

1.5.6. Statistics and random numbers: `scipy.stats`¶

scipy.stats contains fundamental tools for statistics in Python.

1.5.6.1. Statistical Distributions¶

Consider a random variable distributed according to the standard normal. We draw a sample consisting of 100000 observations from the random variable. The normalized histogram of the sample is an estimator of the random variable’s probability density function (PDF):

>>> dist=sp.stats.norm(loc=0,scale=1)# standard normal distribution
>>> sample=dist.rvs(size=100000)# "random variate sample"
>>> plt.hist(sample,bins=50,density=True,label='normalized histogram')
>>> x=np.linspace(-5,5)
>>> plt.plot(x,dist.pdf(x),label='PDF')
[<matplotlib.lines.Line2D object at ...>]
>>> plt.legend()
<matplotlib.legend.Legend object at ...>

../../_images/sphx_glr_plot_normal_distribution_001.png

Suppose we knew that the sample had been drawn from a distribution belonging to the family of normal distributions, but we did not know the particular distribution’s location (mean) and scale (standard deviation). We perform maximum likelihood estimation of the unknown parameters using the distribution family’s fit method:

>>> loc,scale=sp.stats.norm.fit(sample)
>>> loc
np.float64(0.0015767005...)
>>> scale
np.float64(0.9973396878...)

Since we know the true parameters of the distribution from which the sample was drawn, we are not surprised that these estimates are similar.

1.5.6.2. Sample Statistics and Hypothesis Tests¶

The sample mean is an estimator of the mean of the distribution from which the sample was drawn:

>>> np.mean(sample)
np.float64(0.001576700508...)

NumPy includes some of the most fundamental sample statistics (e.g. numpy.mean(), numpy.var(), numpy.percentile()); scipy.stats includes many more. For instance, the geometric mean is a common measure of central tendency for data that tends to be distributed over many orders of magnitude.

>>> sp.stats.gmean(2**sample)
np.float64(1.0010934829...)

SciPy also includes a variety of hypothesis tests that produce a sample statistic and a p-value. For instance, suppose we wish to test the null hypothesis that sample was drawn from a normal distribution:

>>> res=sp.stats.normaltest(sample)
>>> res.statistic
np.float64(5.20841759...)
>>> res.pvalue
np.float64(0.07396163283...)

Here, statistic is a sample statistic that tends to be high for samples that are drawn from non-normal distributions. pvalue is the probability of observing such a high value of the statistic for a sample that has been drawn from a normal distribution. If the p-value is unusually small, this may be taken as evidence that sample was not drawn from the normal distribution. Our statistic and p-value are moderate, so the test is inconclusive.

There are many other features of scipy.stats, including circular statistics, quasi-Monte Carlo methods, and resampling methods. For much more information, see the documentation of scipy.stats and the advanced chapter statistics.

1.5.7. Numerical integration: `scipy.integrate`¶

1.5.7.1. Quadrature¶

Suppose we wish to compute the definite integral $\int_0^{\pi / 2} \sin(t) dt$ numerically. scipy.integrate.quad() chooses one of several adaptive techniques depending on the parameters, and is therefore the recommended first choice for integration of function of a single variable:

>>> integral,error_estimate=sp.integrate.quad(np.sin,0,np.pi/2)
>>> np.allclose(integral,1)# numerical result ~ analytical result
True
>>> abs(integral-1)<error_estimate#  actual error < estimated error
True

Other functions for numerical quadrature, including integration of multivariate functions and approximating integrals from samples, are available in scipy.integrate.

1.5.7.2. Initial Value Problems¶

scipy.integrate also features routines for integrating Ordinary Differential Equations (ODE). For example, scipy.integrate.solve_ivp() integrates ODEs of the form:

$\frac{dy}{dt} = f(t, y(t))$

from an initial time $t_0$ and initial state $y(t=t_0)=t_0$ to a final time $t_f$ or until an event occurs (e.g. a specified state is reached).

As an introduction, consider the initial value problem given by $\frac{dy}{dt} = -2 y$ and the initial condition $y(t=0) = 1$ on the interval $t = 0 \dots 4$ . We begin by defining a callable that computes $f(t, y(t))$ given the current time and state.

>>> deff(t,y):
... return-2*y

Then, to compute y as a function of time:

>>> t_span=(0,4)# time interval
>>> t_eval=np.linspace(*t_span)# times at which to evaluate `y`
>>> y0=[1,]# initial state
>>> res=sp.integrate.solve_ivp(f,t_span=t_span,y0=y0,t_eval=t_eval)

and plot the result:

>>> plt.plot(res.t,res.y[0])
[<matplotlib.lines.Line2D object at ...>]
>>> plt.xlabel('t')
Text(0.5, ..., 't')
>>> plt.ylabel('y')
Text(..., 0.5, 'y')
>>> plt.title('Solution of Initial Value Problem')
Text(0.5, 1.0, 'Solution of Initial Value Problem')

../../_images/sphx_glr_plot_solve_ivp_simple_001.png

Let us integrate a more complex ODE: a damped spring-mass oscillator. The position of a mass attached to a spring obeys the 2nd order ODE $\ddot{y} + 2 \zeta \omega_0 \dot{y} + \omega_0^2 y = 0$ with natural frequency $\omega_0 = \sqrt{k/m}$ , damping ratio $\zeta = c/(2 m \omega_0)$ , spring constant $k$ , mass $m$ , and damping coefficient $c$ .

Before using scipy.integrate.solve_ivp(), the 2nd order ODE needs to be transformed into a system of first-order ODEs. Note that

$\frac{dy}{dt} = \dot{y} \frac{d\dot{y}}{dt} = \ddot{y} = -(2 \zeta \omega_0 \dot{y} + \omega_0^2 y)$

If we define $z = [z_0, z_1]$ where $z_0 = y$ and $z_1 = \dot{y}$ , then the first order equation:

$\frac{dz}{dt} = \begin{bmatrix} \frac{dz_0}{dt} \\ \frac{dz_1}{dt} \end{bmatrix} = \begin{bmatrix} z_1 \\ -(2 \zeta \omega_0 z_1 + \omega_0^2 z_0) \end{bmatrix}$

is equivalent to the original second order equation.

We set:

>>> m=0.5# kg
>>> k=4# N/m
>>> c=0.4# N s/m
>>> zeta=c/(2*m*np.sqrt(k/m))
>>> omega=np.sqrt(k/m)

and define the function that computes $\dot{z} = f(t, z(t))$ :

>>> deff(t,z,zeta,omega):
... return(z[1],-2.0*zeta*omega*z[1]-omega**2*z[0])

../../_images/sphx_glr_plot_solve_ivp_damped_spring_mass_001.png

Integration of the system follows:

>>> t_span=(0,10)
>>> t_eval=np.linspace(*t_span,100)
>>> z0=[1,0]
>>> res=sp.integrate.solve_ivp(f,t_span,z0,t_eval=t_eval,
... args=(zeta,omega),method='LSODA')

Tip

With the option method=’LSODA’, scipy.integrate.solve_ivp() uses the LSODA (Livermore Solver for Ordinary Differential equations with Automatic method switching for stiff and non-stiff problems). See the ODEPACK Fortran library for more details.

1.5.8. Fast Fourier transforms: `scipy.fft`¶

The scipy.fft module computes fast Fourier transforms (FFTs) and offers utilities to handle them. Some important functions are:

scipy.fft.fft() to compute the FFT
scipy.fft.fftfreq() to generate the sampling frequencies
scipy.fft.ifft() to compute the inverse FFT, from frequency space to signal space

As an illustration, a (noisy) input signal (sig), and its FFT:

>>> sig_fft=sp.fft.fft(sig)
>>> freqs=sp.fft.fftfreq(sig.size,d=time_step)


Signal	FFT

As the signal comes from a real-valued function, the Fourier transform is symmetric.

The peak signal frequency can be found with freqs[power.argmax()]

../../_images/sphx_glr_plot_fftpack_003.png

Setting the Fourier component above this frequency to zero and inverting the FFT with scipy.fft.ifft(), gives a filtered signal.

Note

The code of this example can be found here

Fully worked examples:

Crude periodicity finding (link)	Gaussian image blur (link)

1.5.9. Signal processing: `scipy.signal`¶

Tip

scipy.signal is for typical signal processing: 1D, regularly-sampled signals.

../../_images/sphx_glr_plot_resample_001.png

Resampling scipy.signal.resample(): resample a signal to n points using FFT.

>>> t=np.linspace(0,5,100)
>>> x=np.sin(t)
>>> x_resampled=sp.signal.resample(x,25)
>>> plt.plot(t,x)
[<matplotlib.lines.Line2D object at ...>]
>>> plt.plot(t[::4],x_resampled,'ko')
[<matplotlib.lines.Line2D object at ...>]

Tip

Notice how on the side of the window the resampling is less accurate and has a rippling effect.

This resampling is different from the interpolation provided by scipy.interpolate as it only applies to regularly sampled data.

../../_images/sphx_glr_plot_detrend_001.png

Detrending scipy.signal.detrend(): remove linear trend from signal:

>>> t=np.linspace(0,5,100)
>>> rng=np.random.default_rng()
>>> x=t+rng.normal(size=100)
>>> x_detrended=sp.signal.detrend(x)
>>> plt.plot(t,x)
[<matplotlib.lines.Line2D object at ...>]
>>> plt.plot(t,x_detrended)
[<matplotlib.lines.Line2D object at ...>]

Filtering: For non-linear filtering, scipy.signal has filtering (median filter scipy.signal.medfilt(), Wiener scipy.signal.wiener()), but we will discuss this in the image section.

Tip

scipy.signal also has a full-blown set of tools for the design of linear filter (finite and infinite response filters), but this is out of the scope of this tutorial.

Spectral analysis: scipy.signal.spectrogram() compute a spectrogram –frequency spectrums over consecutive time windows–, while scipy.signal.welch() comptes a power spectrum density (PSD).

1.5.10. Image manipulation: `scipy.ndimage`¶

scipy.ndimage provides manipulation of n-dimensional arrays as images.

1.5.10.1. Geometrical transformations on images¶

Changing orientation, resolution, ..

>>> importscipyassp
>>> # Load an image
>>> face=sp.datasets.face(gray=True)
>>> # Shift, rotate and zoom it
>>> shifted_face=sp.ndimage.shift(face,(50,50))
>>> shifted_face2=sp.ndimage.shift(face,(50,50),mode='nearest')
>>> rotated_face=sp.ndimage.rotate(face,30)
>>> cropped_face=face[50:-50,50:-50]
>>> zoomed_face=sp.ndimage.zoom(face,2)
>>> zoomed_face.shape
(1536, 2048)

../../_images/sphx_glr_plot_image_transform_001.png

>>> plt.subplot(151)
<Axes: >
>>> plt.imshow(shifted_face,cmap=plt.cm.gray)
<matplotlib.image.AxesImage object at 0x...>
>>> plt.axis('off')
(np.float64(-0.5), np.float64(1023.5), np.float64(767.5), np.float64(-0.5))
>>> # etc.

1.5.10.2. Image filtering¶

Generate a noisy face:

>>> importscipyassp
>>> face=sp.datasets.face(gray=True)
>>> face=face[:512,-512:]# crop out square on right
>>> importnumpyasnp
>>> noisy_face=np.copy(face).astype(float)
>>> rng=np.random.default_rng()
>>> noisy_face+=face.std()*0.5*rng.standard_normal(face.shape)

Apply a variety of filters on it:

>>> blurred_face=sp.ndimage.gaussian_filter(noisy_face,sigma=3)
>>> median_face=sp.ndimage.median_filter(noisy_face,size=5)
>>> wiener_face=sp.signal.wiener(noisy_face,(5,5))

../../_images/sphx_glr_plot_image_filters_001.png

Other filters in scipy.ndimage.filters and scipy.signal can be applied to images.

1.5.10.3. Mathematical morphology¶

Tip

Mathematical morphology stems from set theory. It characterizes and transforms geometrical structures. Binary (black and white) images, in particular, can be transformed using this theory: the sets to be transformed are the sets of neighboring non-zero-valued pixels. The theory was also extended to gray-valued images.

Mathematical-morphology operations use a structuring element in order to modify geometrical structures.

Let us first generate a structuring element:

>>> el=sp.ndimage.generate_binary_structure(2,1)
>>> el
array([[False, True, False],
       [...True, True, True],
       [False, True, False]])
>>> el.astype(int)
array([[0, 1, 0],
       [1, 1, 1],
       [0, 1, 0]])

Erosion scipy.ndimage.binary_erosion()

>>> a=np.zeros((7,7),dtype=int)
>>> a[1:6,2:5]=1
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]])
>>> sp.ndimage.binary_erosion(a).astype(a.dtype)
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]])
>>> # Erosion removes objects smaller than the structure
>>> sp.ndimage.binary_erosion(a,structure=np.ones((5,5))).astype(a.dtype)
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]])

Dilation scipy.ndimage.binary_dilation()

>>> a=np.zeros((5,5))
>>> a[2,2]=1
>>> a
array([[0.,  0.,  0.,  0.,  0.],
       [0.,  0.,  0.,  0.,  0.],
       [0.,  0.,  1.,  0.,  0.],
       [0.,  0.,  0.,  0.,  0.],
       [0.,  0.,  0.,  0.,  0.]])
>>> sp.ndimage.binary_dilation(a).astype(a.dtype)
array([[0.,  0.,  0.,  0.,  0.],
       [0.,  0.,  1.,  0.,  0.],
       [0.,  1.,  1.,  1.,  0.],
       [0.,  0.,  1.,  0.,  0.],
       [0.,  0.,  0.,  0.,  0.]])

Opening scipy.ndimage.binary_opening()

>>> a=np.zeros((5,5),dtype=int)
>>> a[1:4,1:4]=1
>>> a[4,4]=1
>>> a
array([[0, 0, 0, 0, 0],
       [0, 1, 1, 1, 0],
       [0, 1, 1, 1, 0],
       [0, 1, 1, 1, 0],
       [0, 0, 0, 0, 1]])
>>> # Opening removes small objects
>>> sp.ndimage.binary_opening(a,structure=np.ones((3,3))).astype(int)
array([[0, 0, 0, 0, 0],
       [0, 1, 1, 1, 0],
       [0, 1, 1, 1, 0],
       [0, 1, 1, 1, 0],
       [0, 0, 0, 0, 0]])
>>> # Opening can also smooth corners
>>> sp.ndimage.binary_opening(a).astype(int)
array([[0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0],
       [0, 1, 1, 1, 0],
       [0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0]])

Closing: scipy.ndimage.binary_closing()

An opening operation removes small structures, while a closing operation fills small holes. Such operations can therefore be used to “clean” an image.

>>> a=np.zeros((50,50))
>>> a[10:-10,10:-10]=1
>>> rng=np.random.default_rng()
>>> a+=0.25*rng.standard_normal(a.shape)
>>> mask=a>=0.5
>>> opened_mask=sp.ndimage.binary_opening(mask)
>>> closed_mask=sp.ndimage.binary_closing(opened_mask)

../../_images/sphx_glr_plot_mathematical_morpho_001.png

For gray-valued images, eroding (resp. dilating) amounts to replacing a pixel by the minimal (resp. maximal) value among pixels covered by the structuring element centered on the pixel of interest.

>>> a=np.zeros((7,7),dtype=int)
>>> a[1:6,1:6]=3
>>> a[4,4]=2;a[2,3]=1
>>> a
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 3, 3, 3, 3, 3, 0],
       [0, 3, 3, 1, 3, 3, 0],
       [0, 3, 3, 3, 3, 3, 0],
       [0, 3, 3, 3, 2, 3, 0],
       [0, 3, 3, 3, 3, 3, 0],
       [0, 0, 0, 0, 0, 0, 0]])
>>> sp.ndimage.grey_erosion(a,size=(3,3))
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 3, 2, 2, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]])

1.5.10.4. Connected components and measurements on images¶

Let us first generate a nice synthetic binary image.

>>> x,y=np.indices((100,100))
>>> sig=np.sin(2*np.pi*x/50.)*np.sin(2*np.pi*y/50.)*(1+x*y/50.**2)**2
>>> mask=sig>1

../../_images/sphx_glr_plot_connect_measurements_001.png

../../_images/sphx_glr_plot_connect_measurements_002.png

scipy.ndimage.label() assigns a different label to each connected component:

>>> labels,nb=sp.ndimage.label(mask)
>>> nb
8

Now compute measurements on each connected component:

>>> areas=sp.ndimage.sum(mask,labels,range(1,labels.max()+1))
>>> areas# The number of pixels in each connected component
array([190.,   45.,  424.,  278.,  459.,  190.,  549.,  424.])
>>> maxima=sp.ndimage.maximum(sig,labels,range(1,labels.max()+1))
>>> maxima# The maximum signal in each connected component
array([ 1.80238238,   1.13527605,   5.51954079,   2.49611818, 6.71673619,
        1.80238238,  16.76547217,   5.51954079])

../../_images/sphx_glr_plot_connect_measurements_003.png

Extract the 4th connected component, and crop the array around it:

>>> sp.ndimage.find_objects(labels)[3]
(slice(30, 48, None), slice(30, 48, None))
>>> sl=sp.ndimage.find_objects(labels)[3]
>>> importmatplotlib.pyplotasplt
>>> plt.imshow(sig[sl])
<matplotlib.image.AxesImage object at ...>

See the summary exercise on Image processing application: counting bubbles and unmolten grains for a more advanced example.

1.5.11. Summary exercises on scientific computing ¶

The summary exercises use mainly NumPy, SciPy and Matplotlib. They provide some real-life examples of scientific computing with Python. Now that the basics of working with NumPy and SciPy have been introduced, the interested user is invited to try these exercises.

Exercises:

Proposed solutions:

1.5.11.4. Example of solution for the image processing exercise: unmolten grains in glass

1.5.12. Full code examples for the SciPy chapter ¶

Finding the minimum of a smooth function

Resample a signal with scipy.signal.resample

Detrending a signal

Integrating a simple ODE

Normal distribution: histogram and PDF

Integrate the Damped spring-mass oscillator

Comparing 2 sets of samples from Gaussians

Curve fitting

Spectrogram, power spectral density

Demo mathematical morphology

Plot geometrical transformations on images

Demo connected components

Minima and roots of a function

Plot filtering on images

Optimization of a two-parameter function

Plotting and manipulating FFTs for filtering

A demo of 1D interpolation

1.5.12.18. Solutions of the exercises for SciPy¶

Crude periodicity finding

Curve fitting: temperature as a function of month of the year

Simple image blur by convolution with a Gaussian kernel

Image denoising by FFT

Gallery generated by Sphinx-Gallery

1.5. SciPy : high-level scientific computing¶

1.5.1. File input/output: `scipy.io`¶

1.5.2. Special functions: `scipy.special`¶

1.5.3. Linear algebra operations: `scipy.linalg`¶

1.5.4. Interpolation: `scipy.interpolate`¶

1.5.5. Optimization and fit: `scipy.optimize`¶

1.5.5.1. Root Finding¶

1.5.5.2. Curve fitting¶

1.5.5.3. Optimization¶

1.5.6. Statistics and random numbers: `scipy.stats`¶

1.5.6.1. Statistical Distributions¶

1.5.6.2. Sample Statistics and Hypothesis Tests¶

1.5.7. Numerical integration: `scipy.integrate`¶

1.5.7.1. Quadrature¶

1.5.7.2. Initial Value Problems¶

1.5.8. Fast Fourier transforms: `scipy.fft`¶

1.5.9. Signal processing: `scipy.signal`¶

1.5.10. Image manipulation: `scipy.ndimage`¶

1.5.10.1. Geometrical transformations on images¶

1.5.10.2. Image filtering¶

1.5.10.3. Mathematical morphology¶

1.5.10.4. Connected components and measurements on images¶

1.5.11. Summary exercises on scientific computing ¶

1.5.12. Full code examples for the SciPy chapter ¶

1.5.12.18. Solutions of the exercises for SciPy¶

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