我正在尝试编写一个简单的独立程序，该程序使用 CDF 9/7 小波对一维列表执行单级离散小波变换，然后对其进行重构。我只是使用卷积/滤波器组方法来了解它的工作原理。换句话说，将列表与滤波器进行卷积以获得比例系数，将列表与不同的滤波器进行卷积以获得小波系数，但只能从所有其他元素开始执行此操作。然后上采样(即在元素之间添加零)，对小波和比例系数应用滤波器，将它们加在一起，得到原始列表。
我可以让它为 Haar 小波滤波器工作，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。然而，结果列表和原始列表的总和是一样的。
我确定这是卷积中的一个非常愚蠢的错误，但我就是想不通。我已经尝试了一堆卷积排列，比如将过滤器集中在索引“i”上，而不是从它的左边缘开始，但似乎没有任何效果......这可能是那些会造成的错误之一当我弄清楚时，我拍了拍我的头。
代码如下:
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，我从这里的附录中获取了过滤器值:http://www1.cs.columbia.edu/~rso2102/AWR/Files/Overbeck2009AWR.pdf ，但我也看到它们也用在一堆 matlab 示例代码中。
请您参考如下方法:
实际上，我自己通过比较系数，然后使用此提升实现中的代码进行重构来解决了这个问题:
http://www.embl.de/~gpau/misc/dwt97.c
基本上，我 1) 使边界条件对称，而不是周期性的 2) 必须以某些方式抵消卷积(和上采样)才能使其全部对齐。
这是代码，以防其他人遇到问题。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，但至少它有效。这还包括我用来针对该引用进行测试的“开关”，我必须修改 Haar 小波才能使其工作。
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
