信息处理 - 函数的哈尔级数逼近？ - 吾爱随笔录

函数的哈尔级数逼近？

信息处理小波 Python 转变

2022-01-15 11:00:44

我正在尝试编写函数的 Haar 小波逼近。
我是一个完全的新手。

编辑
下面是我的 f 算法的伪代码，。我在谷歌上到处挑选它。 $f: [0,10] \to \mathbb R$ $f(x)=x$

1) , 其中 $N$ $x \in [0, \infty)$

ψ_{N, j} (x) = \frac{1}{\sqrt{N}} \cdot {\begin{cases} 1 & j = 0 \\ + 2^{p / 2} & a \leq x - ⌊ \frac{x - a}{L} ⌋ L < b \\ - 2^{p / 2} & b \leq x - ⌊ \frac{x - a}{L} ⌋ L < c \\ 0 & otherwise \end{cases}

$\psi_{N,j}(x) = \frac{1}{\sqrt{N}} \cdot\begin{cases} 1 & j =0 \\ +2^{p/2} & a \le x - \left \lfloor \frac{x-a}{L}\right\rfloor L< b \\ -2^{p/2} & b \le x - \left \lfloor \frac{x-a}{L} \right\rfloor L < c\\ 0 & \text{otherwise} \end{cases}$

\begin{aligned} a & = \frac{q - 1}{2^{p}}, b = \frac{q - 0.5}{2^{p}}, c = \frac{q}{2^{p}}, L = c - a, \\ p & = ⌊ \log_{2} (j + 1) ⌋, q = j + 1 - p . \end{aligned}

$\begin{align} a &= \frac{q-1}{2^p}, b = \frac{q-0.5}{2^p}, c = \frac{q}{2^p}, L = c-a, \\ p &= \left\lfloor \log_2(j+1) \right\rfloor, q = j +1 - p. \end{align}$

2) Haar 级数系数

{\hat{ψ}}_{N, j} = \int_{0}^{10} f (t) ψ_{N, j} (t) d t

$\hat \psi_{N,j} = \int_0^{10} f(t) \psi_{N,j}(t) dt$

3) 哈尔级数

f (x) \approx \sum_{j = 0}^{N} {\hat{ψ}}_{N, j} \cdot ψ_{N, j} (x)

$f(x) \approx \sum_{j=0}^N \hat \psi_{N,j} \cdot \psi_{N,j}(x)$

4）对于，这给出了图表 $N=4$ 在此处输入图像描述

1个回答

感谢Jan的代码。我完成了我的实施工作。
下面的代码比较：Haar vs Fourier vs Chebyshev。
在此处输入图像描述

from __future__ import division
from mpmath import *

# --------- Haar wavelet approximation of a function
# algorithm from : http://fourier.eng.hmc.edu/e161/lectures/wavelets/node5.html
# implementation only handle [0,1] for the moment: scaling and wavelet fcts need to be periodice

phi = lambda x : (0 <= x < 1) #scaling fct
psi = lambda x : (0 <= x < .5) - (.5 <= x < 1) #wavelet fct
phi_j_k = lambda x, j, k : 2**(j/2) * phi(2**j * x - k)
psi_j_k = lambda x, j, k : 2**(j/2) * psi(2**j * x - k)

def haar(f, interval, level):
    c0 = quadgl(  lambda t : f(t) * phi_j_k(t, 0, 0), interval  )

    coef = []
    for j in xrange(0, level):
        for k in xrange(0, 2**j):
                djk = quadgl(  lambda t: f(t) * psi_j_k(t, j, k), interval  )
                coef.append( (j, k, djk) )

    return c0, coef

def haarval(haar_coef, x):
    c0, coef = haar_coef
    s = c0 * phi_j_k(x, 0, 0)
    for j, k ,djk in coef:
            s += djk * psi_j_k(x, j, k)
    return s

# --------- to plot an Haar wave
interval = [0, 1]
plot([lambda x : phi_j_k(x,1,1)],interval)

# ---------- main
# below is code to compate : Haar vs Fourier vs Chebyshev

nb_coeff = 5
interval = [0, 1] # haar only handle [0,1] for the moment: scaling and wavelet fcts need to be periodice

fct = lambda x : x

haar_coef = haar(fct, interval, nb_coeff)
haar_series_apx = lambda x : haarval(haar_coef, x)

fourier_coef = fourier(fct, interval, nb_coeff)
fourier_series_apx = lambda x: fourierval(fourier_coef, interval, x)
chebyshev_coef = chebyfit(fct, interval, nb_coeff)
chebyshev_series_apx = lambda x : polyval(chebyshev_coef, x)


print 'fourier %d chebyshev %d haar %d' % ( len(fourier_coef[0]) + len(fourier_coef[1]),len(chebyshev_coef), 1 + len(haar_coef[1]))
print 'error:'
print 'fourier', quadgl(  lambda x : abs( fct(x) - fourier_series_apx(x) ), interval  )
print 'chebyshev', quadgl(  lambda x : abs( fct(x) - chebyshev_series_apx(x) ), interval  )
print 'haar', quadgl(  lambda x : abs( fct(x) - haar_series_apx(x) ), interval  )

plot([fct, fourier_series_apx, chebyshev_series_apx, haar_series_apx], interval)

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