# 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.fftpack 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:

>>> import numpy as np
>>> import scipy as sp


## 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.

>>> import scipy as sp
>>> a = np.ones((3, 3))
>>> sp.io.savemat('file.mat', {'a': a})  # savemat expects a dictionary
>>> 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
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 functions of the first kind), and

• elliptic functions, such as scipy.special.ellipj() (Jacobian 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 .

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


However, it is numerically favorable for small , 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)
0.0
>>> sp.special.xlog1py(x, y)
2.5e-18


Many special functions also have “logarithmized” variants. For instance, the gamma function is related to the factorial function by , 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)
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 .

>>> a = sp.special.gamma(500)
>>> b = sp.special.gamma(499)
>>> a, b
(inf, 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 and , 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
499.0000000...


Similarly, suppose we wish to compute the difference . For this, we use scipy.special.logsumexp(), which computes 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
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:

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


Mathematically, the solution of a linear system is , 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 useful for fitting a function from experimental data and thus evaluating points where no reference value exists. The module includes, but not limited to FITPACK Fortran subroutines.

By imagining experimental data close to a sine function:

>>> measured_time = np.linspace(0, 1, 10)
>>> rng = np.random.default_rng()
>>> noise = (rng.random(10)*2 - 1) * 1e-1
>>> measures = np.sin(2 * np.pi * measured_time) + noise


scipy.interpolate has many interpolation methods which need to be chosen based on the data. See the tutorial for some guidelines:

>>> spline = sp.interpolate.CubicSpline(measured_time, measures) Then the result can be evaluated at the time of interest:

>>> interpolation_time = np.linspace(0, 1, 50)
>>> linear_results = spline(interpolation_time)


scipy.interpolate.CloughTocher2DInterpolator is similar to scipy.interpolate.CubicSpline, but for 2-D arrays. See the summary exercise on Maximum wind speed prediction at the Sprogø station for a more advanced spline interpolation example.

## 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.

>>> def f(x):
...     return (x-1)*(x-2)
>>> def df(x):
...     return 2*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


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().

>>> def f(x):
...     # intersection of unit circle and line from origin
...     return [x**2 + x**2 - 1,
...             x - x]
>>> 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).

>>> def f(x):
...     # intersection of unit circle, line from origin, and parabola
...     return [x**2 + x**2 - 1,
...             x - x,
...             x - x**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¶ 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:

>>> def f(x, a, b, c):
...     return a * np.sin(b * x + c) We then use scipy.optimize.curve_fit() to find and :

>>> 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¶ Suppose we wish to minimize the scalar-valued function of a single variable :

>>> def f(x):
...     return x**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 and a global minimizer near , 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)
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, .

>>> def f(x):
...     return (x - 1)**2 + (x - 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 .)

>>> 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 ...> 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
0.0015767005...
>>> scale
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)
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)
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
5.20841759...
>>> res.pvalue
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¶

Suppose we wish to compute the definite integral 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: from an initial time and initial state to a final time or until an event occurs (e.g. a specified state is reached).

As an introduction, consider the initial value problem given by and the initial condition on the interval . We begin by defining a callable that computes given the current time and state.

>>> def f(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)
[<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') 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 with natural frequency , damping ratio , spring constant , mass , and damping coefficient .

Before using scipy.integrate.solve_ivp(), the 2nd order ODE needs to be transformed into a system of first-order ODEs. Note that If we define where and , then the first order equation: 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 :

>>> def f(t, z, zeta, omega):
...     return (z, -2.0 * zeta * omega * z - omega**2 * z) 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.

Partial Differental Equations

There is no Partial Differential Equations (PDE) solver in SciPy. Some Python packages for solving PDE’s are available, such as fipy or SfePy.

## 1.5.8. Fast Fourier transforms: scipy.fftpack¶

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

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

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


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

The peak signal frequency can be found with freqs[power.argmax()] Setting the Fourrier component above this frequency to zero and inverting the FFT with scipy.fftpack.ifft(), gives a filtered signal.

Note

The code of this example can be found here

Fully worked examples:  ## 1.5.9. Signal processing: scipy.signal¶

Tip

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

>>> import scipy as sp

>>> face = sp.datasets.face(gray=True)

>>> # Shift, roate 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) >>> plt.subplot(151)
<Axes: >

>>> plt.imshow(shifted_face, cmap=plt.cm.gray)
<matplotlib.image.AxesImage object at 0x...>

>>> plt.axis('off')
(-0.5, 1023.5, 767.5, -0.5)

>>> # etc.


### 1.5.10.2. Image filtering¶

Generate a noisy face:

>>> import scipy as sp
>>> face = sp.datasets.face(gray=True)
>>> face = face[:512, -512:]  # crop out square on right
>>> import numpy as np
>>> 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)) 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]])

• >>> 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]])

• >>> 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.]])

• >>> 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]])


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

>>> sp.ndimage.find_objects(labels)
(slice(30, 48, None), slice(30, 48, None))
>>> sl = sp.ndimage.find_objects(labels)
>>> import matplotlib.pyplot as plt
>>> 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:

## 1.5.12. Full code examples for the SciPy chapter¶ Finding the minimum of a smooth function

Finding the minimum of a smooth function Resample a signal with scipy.signal.resample

Resample a signal with scipy.signal.resample Detrending a signal

Detrending a signal Integrating a simple ODE

Integrating a simple ODE Normal distribution: histogram and PDF

Normal distribution: histogram and PDF Integrate the Damped spring-mass oscillator

Integrate the Damped spring-mass oscillator Comparing 2 sets of samples from Gaussians

Comparing 2 sets of samples from Gaussians Curve fitting

Curve fitting Spectrogram, power spectral density

Spectrogram, power spectral density A demo of 1D interpolation

A demo of 1D interpolation Demo mathematical morphology

Demo mathematical morphology Plot geometrical transformations on images

Plot geometrical transformations on images Demo connected components

Demo connected components Minima and roots of a function

Minima and roots of a function Plot filtering on images

Plot filtering on images Optimization of a two-parameter function

Optimization of a two-parameter function Plotting and manipulating FFTs for filtering

Plotting and manipulating FFTs for filtering

### 1.5.12.18. Solutions of the exercises for SciPy¶ Crude periodicity finding

Crude periodicity finding Curve fitting: temperature as a function of month of the year

Curve fitting: temperature as a function of month of the year Simple image blur by convolution with a Gaussian kernel

Simple image blur by convolution with a Gaussian kernel Image denoising by FFT

Image denoising by FFT

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