Question

我正在尝试编写一个简单的自包含程序，使用CDF 9/7小波在1D列表上执行单级离散小波变换，然后重构它。我只是使用卷积/滤波器组方法来掌握它是如何工作的。换句话说，使用过滤器对列表进行卷积以获得比例系数，使用不同的过滤器对列表进行卷积以获得小波系数，但仅从每个其他元素开始。然后上采样（即在元素之间添加零），将滤波器应用于小波并缩放系数，将它们加在一起，并获得原始列表。

我可以让这个用于Haar小波滤波器，但是当我尝试使用CDF 9/7滤波器时，它不会产生相同的输入。但是，结果列表和原始列表总和相同。

我确定这在卷积中是一个非常愚蠢的错误，但我无法弄明白。我已经尝试了一系列卷积的排列，比如将过滤器集中在索引“i”上，而不是启动它的左边缘，但似乎没有任何工作......这可能是其中一个错误当我弄清楚的时候，我打了个耳光。

以下是代码：

import random
import math
length = 128
array = list()
row = list()
scaleCoefficients = list()
waveletCoefficients = list()
reconstruction = list()

def upsample(lst, index):
    if (index % 2 == 0):
        return 0.0
    else:
        return lst[index/2]

for i in range(length):
    array.append(random.random())

## CDF 9/7 Wavelet (doesn't work?)
DWTAnalysisLowpass = [.026749, -.016864, -.078223, .266864, .602949, .266864, -.078223, -.016864, .026749]
for i in range(len(DWTAnalysisLowpass)):
    DWTAnalysisLowpass[i] = math.sqrt(2.0) * DWTAnalysisLowpass[i]
DWTAnalysisHighpass = [0.0, .091272, -.057544, -0.591272, 1.115087, -.591272, -.057544, .091272, 0.0]
for i in range(len(DWTAnalysisHighpass)):
    DWTAnalysisHighpass[i] = 1.0/math.sqrt(2.0) * DWTAnalysisHighpass[i]

DWTSynthesisLowpass = [0.0, -.091272, -.057544, 0.591272, 1.115087, .591272, -.057544, -.091272, 0.0]
for i in range(len(DWTSynthesisLowpass)):
    DWTSynthesisLowpass[i] = 1.0/math.sqrt(2.0) * DWTSynthesisLowpass[i]
DWTSynthesisHighpass = [.026749, .016864, -.078223, -.266864, .602949, -.266864, -.078223, .016864, .026749]
for i in range(len(DWTSynthesisHighpass)):
    DWTSynthesisHighpass[i] = math.sqrt(2.0) * DWTSynthesisHighpass[i]

## Haar Wavelet (Works)
## c = 1.0/math.sqrt(2)
## DWTAnalysisLowpass = [c,c]
## DWTAnalysisHighpass = [c, -c]
## DWTSynthesisLowpass = [c, c]
## DWTSynthesisHighpass = [-c, c]


## Do the forward transform - we only need to do it on half the elements
for i in range(0,length,2):
    newVal = 0.0
    ## Convolve the next j elements
    for j in range(len(DWTAnalysisLowpass)):
        index = i + j
        if(index >= length):
            index = index - length

        newVal = newVal + array[index]*DWTAnalysisLowpass[j]

    scaleCoefficients.append(newVal)

    newVal = 0.0
    for j in range(len(DWTAnalysisHighpass)):
        index = i + j
        if(index >= length):
            index = index - length

        newVal = newVal + array[index]*DWTAnalysisHighpass[j]

    waveletCoefficients.append(newVal)

## Do the inverse transform
for i in range(length):
    newVal = 0.0
    for j in range(len(DWTSynthesisHighpass)):
        index = i + j
        if(index >= length):
            index = index - length

        newVal = newVal + upsample(waveletCoefficients, index)*DWTSynthesisHighpass[j]

    for j in range(len(DWTSynthesisLowpass)):
        index = i + j
        if(index >= length):
            index = index - length

        newVal = newVal + upsample(scaleCoefficients, index)*DWTSynthesisLowpass[j]

    reconstruction.append(newVal)

print sum(reconstruction)
print sum(array)
print reconstruction
print array

顺便说一句，我从附录中获取了过滤器值：http://www1.cs.columbia.edu/~rso2102/AWR/Files/Overbeck2009AWR.pdf，但我也看到它们也被用在一堆matlab示例代码中。

Answer 1

实际上我自己通过比较系数，然后重建，用这个提升实现的代码来解决它：

http://www.embl.de/~gpau/misc/dwt97.c

基本上，我 1）使边界条件对称，而不是周期性 2）必须以某种方式抵消卷积（和上采样）以使其全部排列。

以下是其他人遇到问题的代码。我觉得这仍然过于复杂，特别是因为它没有真正记录在任何地方，但至少它是有效的。这还包括我用来测试该参考的“开关”，我不得不修改Haar小波以使其工作。

import random
import math
length = int()
array = list()
row = list()
scaleCoefficients = list()
waveletCoefficients = list()
reconstruction = list()
switch = False

def upsample1(lst, index):
    if (index % 2 == 0):
        return lst[index/2]
    else:
        return 0.0

def upsample2(lst, index):
    if (index % 2 == 0):
        return 0.0
    else:
        return lst[index/2]

## Generate a random list of floating point numbers
if (not switch):
    length = 128
    for i in range(length):
        array.append(random.random())
else:
    length = 32
    for i in range(32):
        array.append(5.0+i+.4*i*i-.02*i*i*i)

## First Part Just Calculates the Filters
## CDF 9/7 Wavelet
DWTAnalysisLowpass = [.026749, -.016864, -.078223, .266864, .602949, .266864, -.078223, -.016864, .026749]
for i in range(len(DWTAnalysisLowpass)):
    DWTAnalysisLowpass[i] = math.sqrt(2.0) * DWTAnalysisLowpass[i]
DWTAnalysisHighpass = [.091272, -.057544, -0.591272, 1.115087, -.591272, -.057544, .091272]
for i in range(len(DWTAnalysisHighpass)):
    DWTAnalysisHighpass[i] = DWTAnalysisHighpass[i]/math.sqrt(2.0)

DWTSynthesisLowpass = [-.091272, -.057544, 0.591272, 1.115087, .591272, -.057544, -.091272]
for i in range(len(DWTSynthesisLowpass)):
    DWTSynthesisLowpass[i] = DWTSynthesisLowpass[i]/math.sqrt(2.0)
DWTSynthesisHighpass = [.026749, .016864, -.078223, -.266864, .602949, -.266864, -.078223, .016864, .026749]
for i in range(len(DWTSynthesisHighpass)):
    DWTSynthesisHighpass[i] = math.sqrt(2.0) * DWTSynthesisHighpass[i]

## Haar Wavelet
## c = 1.0/math.sqrt(2)
## DWTAnalysisLowpass = [c,c]
## DWTAnalysisHighpass = [c, -c]
## DWTSynthesisLowpass = [-c, c]
## DWTSynthesisHighpass = [c, c]

# Do the forward transform. We can skip every other sample since they would
# be removed in the downsampling anyway
for i in range(0,length,2):
    newVal = 0.0
    ## Convolve the next j elements by the low-pass analysis filter
    for j in range(len(DWTAnalysisLowpass)):
        index = i + j - len(DWTAnalysisLowpass)/2
        if(index >= length):
            index = 2*length - index - 2
        elif (index < 0):
            index = -index

        newVal = newVal + array[index]*DWTAnalysisLowpass[j]

    # append the new value to the list of scale coefficients
    scaleCoefficients.append(newVal)

    newVal = 0.0
    # Convolve the next j elements by the high-pass analysis filter
    for j in range(len(DWTAnalysisHighpass)):
        index = i + j - len(DWTAnalysisHighpass)/2 + 1
        if(index >= length):
            index = 2*length - index - 2
        elif (index < 0):
            index = -index

        newVal = newVal + array[index]*DWTAnalysisHighpass[j]

    # append the new value to the list of wavelet coefficients
    waveletCoefficients.append(newVal)

# Do the inverse transform
for i in range(length):
    newVal = 0.0
    # convolve the upsampled wavelet coefficients with the high-pass synthesis filter
    for j in range(len(DWTSynthesisHighpass)):
        index = i + j - len(DWTSynthesisHighpass)/2
        if(index >= length):
            index = 2*length - index - 2
        elif (index < 0):
            index = -index

        newVal = newVal + upsample2(waveletCoefficients, index)*DWTSynthesisHighpass[j]

    # convolve the upsampled scale coefficients with the low-pass synthesis filter, and
    # add it to the previous convolution
    for j in range(len(DWTSynthesisLowpass)):
        index = i + j - len(DWTSynthesisLowpass)/2
        if(index >= length):
            index = 2*length - index - 2
        elif (index < 0):
            index = -index

        newVal = newVal + upsample1(scaleCoefficients, index)*DWTSynthesisLowpass[j]

    reconstruction.append(newVal)

print ("Sums: ")
print sum(reconstruction)
print sum(array)
print ("Original Signal: ")
print array
if (not switch):
    print ("Wavelet Coefficients: ")
    for i in range(len(scaleCoefficients)):
        print ("sc[" + str(i) + "]: " + str(scaleCoefficients[i]))
    for i in range(len(waveletCoefficients)):
        print ("wc[" + str(i) + "]: " + str(waveletCoefficients[i]))
print ("Reconstruction: ")
print reconstruction

CDF 9/7离散小波变换（卷积）

1 个答案: