SciPy on Clarity

1.scipy Intro

Sun, 24 May 2026 00:00:00 +0000

title: "SciPy-数值计算库" date: 2026-05-24T14:11:00Z

1import numpy as np
2import pylab as pl

1import matplotlib as mpl
2mpl.rcParams['font.sans-serif'] = ['SimHei']

SciPy-数值计算库

1import scipy
2scipy.__version__

'1.0.0'

常数和特殊函数

1from scipy import constants as C
2print (C.c) # 真空中的光速
3print (C.h) # 普朗克常数

299792458.0
6.62607004e-34

1C.physical_constants["electron mass"]

(9.10938356e-31, 'kg', 1.1e-38)

1# 1英里等于多少米, 1英寸等于多少米, 1克等于多少千克, 1磅等于多少千克
2print(C.mile)
3print(C.inch)
4print(C.gram)
5print(C.pound)

1609.3439999999998
0.0254
0.001
0.45359236999999997

1import scipy.special as S

1print (1 + 1e-20)
2print (np.log(1+1e-20))
3print (S.log1p(1e-20))

1.0
0.0
1e-20

1m = np.linspace(0.1, 0.9, 4)
2u = np.linspace(-10, 10, 200)
3results = S.ellipj(u[:, None], m[None, :])
4
5print([y.shape for y in results])

[(200, 4), (200, 4), (200, 4), (200, 4)]

1#%figonly=使用广播计算得到的`ellipj()`返回值
2fig, axes = pl.subplots(2, 2, figsize=(12, 4))
3labels = ["$sn$", "$cn$", "$dn$", "$\phi$"]
4for ax, y, label in zip(axes.ravel(), results, labels):
5    ax.plot(u, y)
6    ax.set_ylabel(label)
7    ax.margins(0, 0.1)
8    
9axes[1, 1].legend(["$m={:g}$".format(m_) for m_ in m], loc="best", ncol=2);

10.scipy Spatial

Sun, 24 May 2026 00:00:00 +0000

title: "10.scipy-spatial" date: 2026-05-24T14:11:03Z

1import numpy as np
2import pylab as pl
3from scipy import spatial

1import matplotlib as mpl
2mpl.rcParams['font.sans-serif'] = ['SimHei']

空间算法库-spatial

计算最近旁点

1x = np.sort(np.random.rand(100))
2idx = np.searchsorted(x, 0.5)
3print (x[idx], x[idx - 1]) #距离0.5最近的数是这两个数中的一个

0.5244435681885733 0.4982156075770372

1from scipy import spatial
2np.random.seed(42)
3N = 100
4points = np.random.uniform(-1, 1, (N, 2))
5kd = spatial.cKDTree(points)
6
7targets = np.array([(0, 0), (0.5, 0.5), (-0.5, 0.5), (0.5, -0.5), (-0.5, -0.5)])
8dist, idx = kd.query(targets, 3)

1r = 0.2
2idx2 = kd.query_ball_point(targets, r)
3idx2

array([list([48]), list([37, 78]), list([22, 79, 92]), list([6, 35, 58]),
       list([7, 42, 55, 83])], dtype=object)

1idx3 = kd.query_pairs(0.1) - kd.query_pairs(0.08)
2idx3

{(1, 46),
 (3, 21),
 (3, 82),
 (3, 95),
 (5, 16),
 (9, 30),
 (10, 87),
 (11, 42),
 (11, 97),
 (18, 41),
 (29, 74),
 (32, 51),
 (37, 78),
 (39, 61),
 (41, 61),
 (50, 84),
 (55, 83),
 (73, 81)}

 1#%figonly=用cKDTree寻找近旁点
 2x, y = points.T
 3colors = "r", "b", "g", "y", "k"
 4
 5fig, (ax1, ax2, ax3) = pl.subplots(1, 3, figsize=(12, 4))
 6
 7for ax in ax1, ax2, ax3:
 8    ax.set_aspect("equal")
 9    ax.plot(x, y, "o", markersize=4)
10    
11for ax in ax1, ax2:
12    for i in range(len(targets)):
13        c = colors[i]
14        tx, ty = targets[i]
15        ax.plot([tx], [ty], "*", markersize=10, color=c)
16
17for i in range(len(targets)):
18    nx, ny = points[idx[i]].T
19    ax1.plot(nx, ny, "o", markersize=10, markerfacecolor="None", 
20             markeredgecolor=colors[i], markeredgewidth=1)
21    
22    nx, ny = points[idx2[i]].T
23    ax2.plot(nx, ny, "o", markersize=10, markerfacecolor="None", 
24             markeredgecolor=colors[i], markeredgewidth=1)
25    
26    ax2.add_artist(pl.Circle(targets[i], r, fill=None, linestyle="dashed"))
27    
28for pidx1, pidx2 in idx3:
29    sx, sy = points[pidx1]
30    ex, ey = points[pidx2]
31    ax3.plot([sx, ex], [sy, ey], "r", linewidth=2, alpha=0.6)
32    
33ax1.set_xlabel(u"搜索最近的3个近旁点")
34ax2.set_xlabel(u"搜索距离在0.2之内的所有近旁点")
35ax3.set_xlabel(u"搜索所有距离在0.08到0.1之间的点对");

2.scipy Optimize

Sun, 24 May 2026 00:00:00 +0000

title: "f计算方程组的误差，[1,1,1]是未知数的初始值" date: 2026-05-24T14:11:06Z

1import pylab as pl
2import numpy as np

1import matplotlib as mpl
2mpl.rcParams['font.sans-serif'] = ['SimHei']

拟合与优化-optimize

非线性方程组求解

 1from math import sin, cos
 2from scipy import optimize
 3
 4def f(x): #❶
 5    x0, x1, x2 = x.tolist() #❷
 6    return [
 7        5*x1+3,
 8        4*x0*x0 - 2*sin(x1*x2),
 9        x1*x2 - 1.5
10    ]
11
12# f计算方程组的误差，[1,1,1]是未知数的初始值
13result = optimize.fsolve(f, [1,1,1]) #❸
14print (result)
15print (f(result))

[-0.70622057 -0.6        -2.5       ]
[0.0, -9.126033262418787e-14, 5.329070518200751e-15]

 1def j(x):  #❶
 2    x0, x1, x2 = x.tolist()
 3    return [[0, 5, 0],
 4            [8 * x0, -2 * x2 * cos(x1 * x2), -2 * x1 * cos(x1 * x2)],
 5            [0, x2, x1]]
 6
 7
 8result = optimize.fsolve(f, [1, 1, 1], fprime=j)  #❷
 9print(result)
10print(f(result))

[-0.70622057 -0.6        -2.5       ]
[0.0, -9.126033262418787e-14, 5.329070518200751e-15]

最小二乘拟合

 1import numpy as np
 2from scipy import optimize
 3
 4X = np.array([ 8.19,  2.72,  6.39,  8.71,  4.7 ,  2.66,  3.78])
 5Y = np.array([ 7.01,  2.78,  6.47,  6.71,  4.1 ,  4.23,  4.05])
 6
 7def residuals(p): #❶
 8    "计算以p为参数的直线和原始数据之间的误差"
 9    k, b = p
10    return Y - (k*X + b)
11
12# leastsq使得residuals()的输出数组的平方和最小，参数的初始值为[1,0]
13r = optimize.leastsq(residuals, [1, 0]) #❷
14k, b = r[0]
15print ("k =",k, "b =",b)

k = 0.6134953491930442 b = 1.794092543259387

 1#%figonly=最小化正方形面积之和（左），误差曲面（右）
 2scale_k = 1.0
 3scale_b = 10.0
 4scale_error = 1000.0
 5
 6def S(k, b):
 7    "计算直线y=k*x+b和原始数据X、Y的误差的平方和"
 8    error = np.zeros(k.shape)
 9    for x, y in zip(X, Y):
10        error += (y - (k * x + b)) ** 2
11    return error
12
13ks, bs = np.mgrid[k - scale_k:k + scale_k:40j, b - scale_b:b + scale_b:40j]
14error = S(ks, bs) / scale_error
15
16from mpl_toolkits.mplot3d import Axes3D
17from matplotlib.patches import Rectangle
18
19fig = pl.figure(figsize=(12, 5))
20
21ax1 = pl.subplot(121)
22
23ax1.plot(X, Y, "o")
24X0 = np.linspace(2, 10, 3)
25Y0 = k*X0 + b
26ax1.plot(X0, Y0)
27
28for x, y in zip(X, Y):
29    y2 = k*x+b
30    rect = Rectangle((x,y), abs(y-y2), y2-y, facecolor="red", alpha=0.2)
31    ax1.add_patch(rect)
32
33ax1.set_aspect("equal")
34
35
36ax2 = fig.add_subplot(122, projection='3d')
37
38ax2.plot_surface(
39    ks, bs / scale_b, error, rstride=3, cstride=3, cmap="jet", alpha=0.5)
40ax2.scatter([k], [b / scale_b], [S(k, b) / scale_error], c="r", s=20)
41ax2.set_xlabel("$k$")
42ax2.set_ylabel("$b$")
43ax2.set_zlabel("$error$");

3.scipy Linalg

Sun, 24 May 2026 00:00:00 +0000

title: "3.scipy-linalg" date: 2026-05-24T14:11:09Z

1import pylab as pl
2import numpy as np
3from scipy import linalg

1import matplotlib as mpl
2mpl.rcParams['font.sans-serif'] = ['SimHei']

线性代数-linalg

解线性方程组

 1import numpy as np
 2from scipy import linalg
 3
 4m, n = 500, 50
 5A = np.random.rand(m, m)
 6B = np.random.rand(m, n)
 7X1 = linalg.solve(A, B)
 8X2 = np.dot(linalg.inv(A), B)
 9print (np.allclose(X1, X2))
10%timeit linalg.solve(A, B)
11%timeit np.dot(linalg.inv(A), B)

True
5.38 ms ± 120 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
8.14 ms ± 994 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

1luf = linalg.lu_factor(A)
2X3 = linalg.lu_solve(luf, B)
3np.allclose(X1, X3)

True

 1M, N = 1000, 100
 2np.random.seed(0)
 3A = np.random.rand(M, M)
 4B = np.random.rand(M, N)
 5Ai = linalg.inv(A)
 6luf = linalg.lu_factor(A)   
 7%timeit linalg.inv(A)
 8%timeit np.dot(Ai, B)
 9%timeit linalg.lu_factor(A)    
10%timeit linalg.lu_solve(luf, B)

50.6 ms ± 1.94 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
3.49 ms ± 306 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
20.1 ms ± 1.42 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
4.49 ms ± 65 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

最小二乘解

 1from numpy.lib.stride_tricks import as_strided
 2
 3
 4def make_data(m, n, noise_scale):  #❶
 5    np.random.seed(42)
 6    x = np.random.standard_normal(m)
 7    h = np.random.standard_normal(n)
 8    y = np.convolve(x, h)
 9    yn = y + np.random.standard_normal(len(y)) * noise_scale * np.max(y)
10    return x, yn, h
11
12
13def solve_h(x, y, n):  #❷
14    X = as_strided(
15        x, shape=(len(x) - n + 1, n), strides=(x.itemsize, x.itemsize))  #❸
16    Y = y[n - 1:len(x)]  #❹
17    h = linalg.lstsq(X, Y)  #❺
18    return h[0][::-1]  #❻

1x, yn, h = make_data(1000, 100, 0.4)
2H1 = solve_h(x, yn, 120)
3H2 = solve_h(x, yn, 80)
4
5print("Average error of H1:", np.mean(np.abs(h[:100] - h)))
6print("Average error of H2:", np.mean(np.abs(h[:80] - H2)))

Average error of H1: 0.0
Average error of H2: 0.2958422158342371

 1#%figonly=实际的系统参数与最小二乘解的比较
 2fig, (ax1, ax2) = pl.subplots(2, 1, figsize=(6, 4))
 3ax1.plot(h, linewidth=2, label=u"实际的系统参数")
 4ax1.plot(H1, linewidth=2, label=u"最小二乘解H1", alpha=0.7)
 5ax1.legend(loc="best", ncol=2)
 6ax1.set_xlim(0, len(H1))
 7
 8ax2.plot(h, linewidth=2, label=u"实际的系统参数")
 9ax2.plot(H2, linewidth=2, label=u"最小二乘解H2", alpha=0.7)
10ax2.legend(loc="best", ncol=2)
11ax2.set_xlim(0, len(H1));

4.scipy Stats

Sun, 24 May 2026 00:00:00 +0000

title: "4.scipy-stats" date: 2026-05-24T14:11:12Z

1import numpy as np
2import pylab as pl
3from scipy import stats

1import matplotlib.pyplot as plt
2import matplotlib as mpl
3mpl.rcParams['font.sans-serif'] = ['SimHei']

统计-stats

连续概率分布

1from scipy import stats
2[k for k, v in stats.__dict__.items() if isinstance(v, stats.rv_continuous)]

['ksone',
 'kstwobign',
 'norm',
 'alpha',
 'anglit',
 'arcsine',
 'beta',
 'betaprime',
 'bradford',
 'burr',
 'burr12',
 'fisk',
 'cauchy',
 'chi',
 'chi2',
 'cosine',
 'dgamma',
 'dweibull',
 'expon',
 'exponnorm',
 'exponweib',
 'exponpow',
 'fatiguelife',
 'foldcauchy',
 'f',
 'foldnorm',
 'weibull_min',
 'weibull_max',
 'frechet_r',
 'frechet_l',
 'genlogistic',
 'genpareto',
 'genexpon',
 'genextreme',
 'gamma',
 'erlang',
 'gengamma',
 'genhalflogistic',
 'gompertz',
 'gumbel_r',
 'gumbel_l',
 'halfcauchy',
 'halflogistic',
 'halfnorm',
 'hypsecant',
 'gausshyper',
 'invgamma',
 'invgauss',
 'invweibull',
 'johnsonsb',
 'johnsonsu',
 'laplace',
 'levy',
 'levy_l',
 'levy_stable',
 'logistic',
 'loggamma',
 'loglaplace',
 'lognorm',
 'gilbrat',
 'maxwell',
 'mielke',
 'kappa4',
 'kappa3',
 'nakagami',
 'ncx2',
 'ncf',
 't',
 'nct',
 'pareto',
 'lomax',
 'pearson3',
 'powerlaw',
 'powerlognorm',
 'powernorm',
 'rdist',
 'rayleigh',
 'reciprocal',
 'rice',
 'recipinvgauss',
 'semicircular',
 'skewnorm',
 'trapz',
 'triang',
 'truncexpon',
 'truncnorm',
 'tukeylambda',
 'uniform',
 'vonmises',
 'vonmises_line',
 'wald',
 'wrapcauchy',
 'gennorm',
 'halfgennorm',
 'crystalball',
 'argus']

1stats.norm.stats()

(array(0.), array(1.))

1X = stats.norm(loc=1.0, scale=2.0)
2X.stats()

(array(1.), array(4.))

1x = X.rvs(size=10000) # 对随机变量取10000个值
2np.mean(x), np.var(x) # 期望值和方差

(1.0048352738823323, 3.9372117720073554)

1stats.norm.fit(x) # 得到随机序列期望值和标准差

(1.0048352738823323, 1.984240855341749)

1pdf, t = np.histogram(x, bins=100, normed=True)  #❶
2t = (t[:-1] + t[1:]) * 0.5  #❷
3cdf = np.cumsum(pdf) * (t[1] - t[0])  #❸
4p_error = pdf - X.pdf(t)
5c_error = cdf - X.cdf(t)
6print ("max pdf error: {}, max cdf error: {}".format(
7    np.abs(p_error).max(),
8    np.abs(c_error).max()))

max pdf error: 0.018998755595167102, max cdf error: 0.018503342378306975


C:\ProgramData\Anaconda3\lib\site-packages\ipykernel_launcher.py:1: VisibleDeprecationWarning: Passing `normed=True` on non-uniform bins has always been broken, and computes neither the probability density function nor the probability mass function. The result is only correct if the bins are uniform, when density=True will produce the same result anyway. The argument will be removed in a future version of numpy.
  """Entry point for launching an IPython kernel.

1#%figonly=正态分布的概率密度函数（左）和累积分布函数（右）
2fig, (ax1, ax2) = pl.subplots(1, 2, figsize=(7, 2))
3ax1.plot(t, pdf, label=u"统计值")
4ax1.plot(t, X.pdf(t), label=u"理论值", alpha=0.6)
5ax1.legend(loc="best")
6ax2.plot(t, cdf)
7ax2.plot(t, X.cdf(t), alpha=0.6);

5.scipy Integrate

Sun, 24 May 2026 00:00:00 +0000

title: "调用ode对lorenz进行求解, 用两个不同的初始值" date: 2026-05-24T14:11:14Z

1import pylab as pl
2import numpy as np
3from scipy import integrate
4from scipy.integrate import odeint
5import matplotlib as mpl
6mpl.rcParams['font.sans-serif'] = ['SimHei']

数值积分-integrate

球的体积

1def half_circle(x):
2    return (1-x**2)**0.5

1N = 10000
2x = np.linspace(-1, 1, N)
3dx = x[1] - x[0]
4y = half_circle(x)
52 * dx * np.sum(y) # 面积的两倍

3.1415893269307373

1np.trapz(y, x) * 2 # 面积的两倍

3.1415893269315975

1from scipy import integrate
2pi_half, err = integrate.quad(half_circle, -1, 1)
3pi_half * 2

3.141592653589797

1def half_sphere(x, y):
2    return (1-x**2-y**2)**0.5

1volume, error = integrate.dblquad(half_sphere, -1, 1, 
2        lambda x:-half_circle(x), 
3        lambda x:half_circle(x))
4
5print (volume, error, np.pi*4/3/2)

2.094395102393199 1.0002356720661965e-09 2.0943951023931953

解常微分方程组

 1#%fig=洛伦茨吸引子：微小的初值差别也会显著地影响运动轨迹
 2from scipy.integrate import odeint 
 3import numpy as np 
 4
 5def lorenz(w, t, p, r, b): #❶
 6    # 给出位置矢量w，和三个参数p, r, b计算出
 7    # dx/dt, dy/dt, dz/dt的值
 8    x, y, z = w.tolist()
 9    # 直接与lorenz的计算公式对应 
10    return p*(y-x), x*(r-z)-y, x*y-b*z
11
12t = np.arange(0, 30, 0.02) # 创建时间点 
13# 调用ode对lorenz进行求解, 用两个不同的初始值 
14track1 = odeint(lorenz, (0.0, 1.00, 0.0), t, args=(10.0, 28.0, 3.0)) #❷
15track2 = odeint(lorenz, (0.0, 1.01, 0.0), t, args=(10.0, 28.0, 3.0)) #❸
16#%hide
17from mpl_toolkits.mplot3d import Axes3D
18fig = pl.figure()
19ax = Axes3D(fig)
20ax.plot(track1[:,0], track1[:,1], track1[:,2], lw=1)
21ax.plot(track2[:,0], track2[:,1], track2[:,2], lw=1);

6.scipy Signal

Sun, 24 May 2026 00:00:00 +0000

title: "设计一个带通滤波器：" date: 2026-05-24T14:11:17Z

1import pylab as pl
2import numpy as np
3from scipy import signal

1import matplotlib as mpl
2mpl.rcParams['font.sans-serif'] = ['SimHei']

信号处理-signal

中值滤波

 1#%fig=使用中值滤波剔除瞬间噪声
 2t = np.arange(0, 20, 0.1)
 3x = np.sin(t)
 4x[np.random.randint(0, len(t), 20)] += np.random.standard_normal(20)*0.6 #❶
 5x2 = signal.medfilt(x, 5) #❷
 6x3 = signal.order_filter(x, np.ones(5), 2)
 7print (np.all(x2 == x3))
 8pl.plot(t, x, label=u"带噪声的信号")
 9pl.plot(t, x2 + 0.5, alpha=0.6, label=u"中值滤波之后的信号")
10pl.legend(loc="best");

True

滤波器设计

 1sampling_rate = 8000.0
 2
 3# 设计一个带通滤波器：
 4# 通带为0.2*4000 - 0.5*4000
 5# 阻带为<0.1*4000, >0.6*4000
 6# 通带增益的最大衰减值为2dB
 7# 阻带的最小衰减值为40dB
 8b, a = signal.iirdesign([0.2, 0.5], [0.1, 0.6], 2, 40) #❶
 9
10# 使用freq计算滤波器的频率响应
11w, h = signal.freqz(b, a) #❷
12
13# 计算增益
14power = 20*np.log10(np.clip(np.abs(h), 1e-8, 1e100)) #❸
15freq = w / np.pi * sampling_rate / 2

 1#%fig=用频率扫描波测量的频率响应
 2# 产生2秒钟的取样频率为sampling_rate Hz的频率扫描信号
 3# 开始频率为0， 结束频率为sampling_rate/2
 4t = np.arange(0, 2, 1/sampling_rate) #❶
 5sweep = signal.chirp(t, f0=0, t1=2, f1=sampling_rate/2) #❷
 6# 对频率扫描信号进行滤波
 7out = signal.lfilter(b, a, sweep) #❸
 8# 将波形转换为能量
 9out = 20*np.log10(np.abs(out)) #❹
10# 找到所有局部最大值的下标
11index = signal.argrelmax(out, order=3)  #❺
12# 绘制滤波之后的波形的增益
13pl.figure(figsize=(8, 2.5))
14pl.plot(freq, power, label=u"带通IIR滤波器的频率响应") 
15pl.plot(t[index]/2.0*4000, out[index], label=u"频率扫描波测量的频谱", alpha=0.6) #❻
16pl.legend(loc="best")
17#%hide
18pl.title(u"频率扫描波测量的滤波器频谱")
19pl.ylim(-100,20)
20pl.ylabel(u"增益(dB)")
21pl.xlabel(u"频率(Hz)");

7.scipy Interpolate

Sun, 24 May 2026 00:00:00 +0000

title: "X-Y轴分为15*15的网格" date: 2026-05-24T14:11:20Z

1import numpy as np
2import pylab as pl
3from scipy import interpolate

1import matplotlib as mpl
2mpl.rcParams['font.sans-serif'] = ['SimHei']

插值-interpolate

一维插值

WARNING

高次interp1d()插值的运算量很大，因此对于点数较多的数据，建议使用后面介绍的UnivariateSpline()。

 1#%fig=`interp1d`的各阶插值
 2from scipy import interpolate
 3
 4x = np.linspace(0, 10, 11)
 5y = np.sin(x)
 6
 7xnew = np.linspace(0, 10, 101)
 8pl.plot(x, y, 'ro')
 9for kind in ['nearest', 'zero', 'slinear', 'quadratic']:
10    f = interpolate.interp1d(x, y, kind=kind)  #❶
11    ynew = f(xnew)  #❷
12    pl.plot(xnew, ynew, label=str(kind))
13
14pl.legend(loc='lower right')

<matplotlib.legend.Legend at 0x274ccb5f6a0>

外推和Spline拟合

 1#%fig=使用UnivariateSpline进行插值：外推（上），数据拟合（下）
 2x1 = np.linspace(0, 10, 20)
 3y1 = np.sin(x1)
 4sx1 = np.linspace(0, 12, 100)
 5sy1 = interpolate.UnivariateSpline(x1, y1, s=0)(sx1)  #❶
 6
 7x2 = np.linspace(0, 20, 200)
 8y2 = np.sin(x2) + np.random.standard_normal(len(x2)) * 0.2
 9sx2 = np.linspace(0, 20, 2000)
10spline2 = interpolate.UnivariateSpline(x2, y2, s=8)  #❷
11sy2 = spline2(sx2)
12
13pl.figure(figsize=(8, 5))
14pl.subplot(211)
15pl.plot(x1, y1, ".", label=u"数据点")
16pl.plot(sx1, sy1, label=u"spline曲线")
17pl.legend()
18
19pl.subplot(212)
20pl.plot(x2, y2, ".", label=u"数据点")
21pl.plot(sx2, sy2, linewidth=2, label=u"spline曲线")
22pl.plot(x2, np.sin(x2), label=u"无噪声曲线")
23pl.legend()

<matplotlib.legend.Legend at 0x274bb27f588>

8.scipy Sparse

Sun, 24 May 2026 00:00:00 +0000

title: "8.scipy-sparse" date: 2026-05-24T14:11:23Z

1%matplotlib inline
2import numpy as np
3import pylab as pl
4from scipy import sparse
5from scipy.sparse import csgraph

稀疏矩阵-sparse

稀疏矩阵的储存形式

1from scipy import sparse
2a = sparse.dok_matrix((10, 5))
3
4a[2, 3] = 1.0
5a[3, 3] = 2.0
6a[4, 3] = 3.0
7print(a.keys())
8print(a.values())

dict_keys([(2, 3), (3, 3), (4, 3)])
dict_values([1.0, 2.0, 3.0])

1b = sparse.lil_matrix((10, 5))
2b[2, 3] = 1.0
3b[3, 4] = 2.0
4b[3, 2] = 3.0
5print(b.data)
6print(b.rows)

[list([]) list([]) list([1.0]) list([3.0, 2.0]) list([]) list([]) list([])
 list([]) list([]) list([])]
[list([]) list([]) list([3]) list([2, 4]) list([]) list([]) list([])
 list([]) list([]) list([])]

1row = [2, 3, 3, 2]
2col = [3, 4, 2, 3]
3data = [1, 2, 3, 10]
4c = sparse.coo_matrix((data, (row, col)), shape=(5, 6))
5print (c.col, c.row, c.data)
6print (c.toarray())

[3 4 2 3] [2 3 3 2] [ 1  2  3 10]
[[ 0  0  0  0  0  0]
 [ 0  0  0  0  0  0]
 [ 0  0  0 11  0  0]
 [ 0  0  3  0  2  0]
 [ 0  0  0  0  0  0]]

矩阵向量相乘

1 import numpy as np
2from scipy.sparse import csr_matrix
3A = csr_matrix([[1, 2, 0], [0, 0, 3], [4, 0, 5]])
4v = np.array([1, 0, -1])
5A.dot(v)

array([ 1, -3, -1], dtype=int32)

示例1

构造一个1000x1000 lil_matrix并添加值：

9.scipy Ndimage

Sun, 24 May 2026 00:00:00 +0000

title: "9.scipy-ndimage" date: 2026-05-24T14:11:26Z

1import numpy as np
2import pylab as pl

图像处理-ndimage

形态学图像处理

 1import numpy as np
 2
 3def expand_image(img, value, out=None, size = 10):
 4    if out is None:
 5        w, h = img.shape
 6        out = np.zeros((w*size, h*size),dtype=np.uint8)
 7    
 8    tmp = np.repeat(np.repeat(img,size,0),size,1)
 9    out[:,:] = np.where(tmp, value, out)
10    out[::size,:] = 0
11    out[:,::size] = 0
12    return out
13    
14def show_image(*imgs): 
15    for idx, img in enumerate(imgs, 1):
16        ax = pl.subplot(1, len(imgs), idx)
17        pl.imshow(img, cmap="gray")   
18        ax.set_axis_off()    
19    pl.subplots_adjust(0.02, 0, 0.98, 1, 0.02, 0)

膨胀和腐蚀

 1#%fig=四连通和八连通的膨胀运算
 2from scipy.ndimage import morphology
 3
 4def dilation_demo(a, structure=None):
 5    b = morphology.binary_dilation(a, structure)
 6    img = expand_image(a, 255)
 7    return expand_image(np.logical_xor(a,b), 150, out=img)
 8
 9a = pl.imread("scipy_morphology_demo.png")[:,:,0].astype(np.uint8)
10img1 = expand_image(a, 255)
11
12img2 = dilation_demo(a)
13img3 = dilation_demo(a, [[1,1,1],[1,1,1],[1,1,1]])
14show_image(img1, img2, img3)

1#%fig=不同结构元素的膨胀效果
2img4 = dilation_demo(a, [[0,0,0],[1,1,1],[0,0,0]])
3img5 = dilation_demo(a, [[0,1,0],[0,1,0],[0,1,0]])
4img6 = dilation_demo(a, [[0,1,0],[0,1,0],[0,0,0]])
5show_image(img4, img5, img6)

04.01 Scientific Python 简介

Tue, 05 May 2026 00:00:00 +0000

SciPy科学计算库简介

04.02 插值

Tue, 05 May 2026 00:00:00 +0000

SciPy中的插值方法

04.03 概率统计方法

Tue, 05 May 2026 00:00:00 +0000

SciPy中的概率统计方法

04.04 曲线拟合

Tue, 05 May 2026 00:00:00 +0000

SciPy中的曲线拟合方法

04.05 最小化函数

Tue, 05 May 2026 00:00:00 +0000

SciPy中的最小化函数

04.06 积分

Tue, 05 May 2026 00:00:00 +0000

SciPy中的积分计算

04.07 解微分方程

Tue, 05 May 2026 00:00:00 +0000

SciPy中解常微分方程的方法

04.08 稀疏矩阵

Tue, 05 May 2026 00:00:00 +0000

SciPy中稀疏矩阵的使用

04.09 线性代数

Tue, 05 May 2026 00:00:00 +0000

SciPy中的线性代数计算

04.10 稀疏矩阵的线性代数

Tue, 05 May 2026 00:00:00 +0000

SciPy中稀疏矩阵的线性代数操作