python - 为什么在填充输入之一时python scipy互相关不起作用
问题描述
scipy 互相关函数根本不适用于特定的一维数组，我不知道为什么。下面的代码演示了这个问题，只需使用一个跟踪而不是另一个跟踪即可。
这个问题有点与互相关和Python互相关没有返回正确的移位有关
#!/usr/bin/python3
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
def _main():
"""
trace = np.array([0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, # down the step
0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, # up the step
0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002]) # down the step
"""
trace = np.array([0.51231204949426460, 0.47472182808002383, 0.48806029762272723, 0.51352464310119930, 0.58506742537603330, 0.62993314829830390, 0.57657927012749040, 0.55369158834668990, 0.56255864527226200, 0.61576098682569510,
0.62955418648769630, 0.64236215760241170, 0.69063835641941580, 0.75073729780384960, 0.86896478361172370, 0.92216712516515690, 0.91329988783884970, 0.92807831604813670, 0.99113300320800610, 0.99999999999999990, 0.91527040506699960,
0.80098377331469030, 0.71723934679539750, 0.68275634764039450, 0.65812563395824950, 0.63250963159524040, 0.59999708953480900, 0.55172083058422660, 0.54975037348965490, 0.57011178351142090, 0.52807534544936740])
left_padded_trace = np.pad(trace, (10, 0), mode='constant', constant_values=trace.min())
center_padded_trace = np.pad(trace, (5, 5), mode='constant', constant_values=trace.min())
right_padded_trace = np.pad(trace, (0, 10), mode='constant', constant_values=trace.min())
correlation1 = signal.correlate(center_padded_trace, left_padded_trace, mode='full', method='fft')
correlation2 = signal.correlate(center_padded_trace, center_padded_trace, mode='full', method='fft')
correlation3 = signal.correlate(center_padded_trace, right_padded_trace, mode='full', method='fft')
corr_peak_index1 = np.argmax(correlation1)
corr_max1 = np.max(correlation1)
corr_peak_index2 = np.argmax(correlation2)
corr_max2 = np.max(correlation2)
corr_peak_index3 = np.argmax(correlation3)
corr_max3 = np.max(correlation3)
offset1 = corr_peak_index1-(center_padded_trace.size-1)
offset2 = corr_peak_index2-(center_padded_trace.size-1)
offset3 = corr_peak_index3-(center_padded_trace.size-1)
print("Corr1: {}, Corr2: {}, Corr3: {}".format(corr_peak_index1, corr_peak_index2, corr_peak_index3))
print("Offset1: {}, Offset2: {}, Offset3: {}".format(offset1, offset2, offset3))
plt.figure(1)
plt.subplot(311)
plt.plot(range(0, center_padded_trace.size), center_padded_trace, 'r-',
range(offset1, left_padded_trace.size+offset1), left_padded_trace, 'b--',
range(0, correlation1.size), correlation1/corr_max1, 'g-',
[corr_peak_index1], [1], 'k+')
plt.subplot(312)
plt.plot(range(0, center_padded_trace.size), center_padded_trace, 'r-',
range(offset2, center_padded_trace.size+offset2), center_padded_trace, 'b--',
range(0, correlation2.size), correlation2/corr_max2, 'g-',
[corr_peak_index2], [1], 'k+')
plt.subplot(313)
plt.plot(range(0, center_padded_trace.size), center_padded_trace, 'r-',
range(offset3, right_padded_trace.size+offset3), right_padded_trace, 'b--',
range(0, correlation3.size), correlation3/corr_max3, 'g-',
[corr_peak_index3], [1], 'k+')
plt.show()
由于填充添加的移位是相同的，唯一的区别是输入轨迹上的变化，因此相关性的移位和对齐结果应该是相同的，但事实并非如此。
对于第一个跟踪（更多合成步骤），相关性和偏移量是：（1 是左侧填充，2 是居中，3 是右侧填充）
- 校正 1：35，校正 2：40，校正 3：45
- 偏移 1：-5，偏移 2：0，偏移 3：5
对于第二个轨迹（更自然），
- 校正 1：40，校正 2：40，校正 3：40
- 偏移1：0，偏移2：0，偏移3：0
跟随情节：
绘制合成迹线（代码中的第一个）
绘制自然轨迹（代码中的第二个）
解决方案
请参阅下面 Paul Panzer 的回答和评论。
问题在于原始代码与非零填充有关。
当用非零值移动数组时，互相关值越来越高，峰值受到影响。以下代码和图像演示了这种效果：
trace = np.array([0.51231204949426460, 0.47472182808002383, 0.48806029762272723, 0.51352464310119930, 0.58506742537603330, 0.62993314829830390, 0.57657927012749040, 0.55369158834668990, 0.56255864527226200, 0.61576098682569510, 0.62955418648769630, 0.64236215760241170, 0.69063835641941580, 0.75073729780384960, 0.86896478361172370, 0.92216712516515690, 0.91329988783884970, 0.92807831604813670, 0.99113300320800610, 0.99999999999999990, 0.91527040506699960, 0.80098377331469030, 0.71723934679539750, 0.68275634764039450, 0.65812563395824950, 0.63250963159524040, 0.59999708953480900, 0.55172083058422660, 0.54975037348965490, 0.57011178351142090, 0.52807534544936740])
for padding_value in np.arange(0, trace.min(), trace.min()/10):
left_padded_trace = np.pad(trace, (10, 0), mode='constant', constant_values=padding_value)
center_padded_trace = np.pad(trace, (5, 5), mode='constant', constant_values=padding_value)
correlation = signal.correlate(center_padded_trace, left_padded_trace, mode='full', method='fft')
corr_peak_index = np.argmax(correlation)
plt.figure(2)
plt.subplot(211)
plt.title('Left Padded Trace')
plt.xticks([])
plt.plot(left_padded_trace)
plt.subplot(212)
plt.title('Centered Padded Trace')
plt.plot(center_padded_trace)
plt.figure(3)
plt.plot(range(0, correlation.size), correlation)
plt.plot([corr_peak_index], [correlation[corr_peak_index]], 'k+')
plt.show()
结果如下所示。可以看出，随着填充值的增加，相关峰值向中心移动。
具有不同填充级别的跟踪，从 0 到最小值
- 相关值和峰值
- 相关值和峰值
解决方案
差异可以通过以下事实来解释，即在第二次跟踪的情况下，您正在填充不为零的最小值。因此，您不能期望峰值会随着偏移量而移动。相反，您会得到移动的峰值曲线加上一个与最小值成比例的三角形。
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
def _main(offset=0, trace_idx=0):
trace = [np.array([0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, # down the step
0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, 0.99999999999999999, 0.99999999999999998, # up the step
0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002, 0.00000000000000001, 0.00000000000000002]), # down the step
np.array([0.51231204949426460, 0.47472182808002383, 0.48806029762272723, 0.51352464310119930, 0.58506742537603330, 0.62993314829830390, 0.57657927012749040, 0.55369158834668990, 0.56255864527226200, 0.61576098682569510,
0.62955418648769630, 0.64236215760241170, 0.69063835641941580, 0.75073729780384960, 0.86896478361172370, 0.92216712516515690, 0.91329988783884970, 0.92807831604813670, 0.99113300320800610, 0.99999999999999990, 0.91527040506699960,
0.80098377331469030, 0.71723934679539750, 0.68275634764039450, 0.65812563395824950, 0.63250963159524040, 0.59999708953480900, 0.55172083058422660, 0.54975037348965490, 0.57011178351142090, 0.52807534544936740])][trace_idx]
trace += offset - trace.min()
left_padded_trace = np.pad(trace, (10, 0), mode='constant', constant_values=trace.min())
center_padded_trace = np.pad(trace, (5, 5), mode='constant', constant_values=trace.min())
right_padded_trace = np.pad(trace, (0, 10), mode='constant', constant_values=trace.min())
correlation1 = signal.correlate(center_padded_trace, left_padded_trace, mode='full', method='fft')
correlation2 = signal.correlate(center_padded_trace, center_padded_trace, mode='full', method='fft')
correlation3 = signal.correlate(center_padded_trace, right_padded_trace, mode='full', method='fft')
corr_peak_index1 = np.argmax(correlation1)
corr_max1 = np.max(correlation1)
corr_peak_index2 = np.argmax(correlation2)
corr_max2 = np.max(correlation2)
corr_peak_index3 = np.argmax(correlation3)
corr_max3 = np.max(correlation3)
offset1 = corr_peak_index1-(center_padded_trace.size-1)
offset2 = corr_peak_index2-(center_padded_trace.size-1)
offset3 = corr_peak_index3-(center_padded_trace.size-1)
return offset1, offset2, offset3
print("Corr1: {}, Corr2: {}, Corr3: {}".format(corr_peak_index1, corr_peak_index2, corr_peak_index3))
print("Offset1: {}, Offset2: {}, Offset3: {}".format(offset1, offset2, offset3))
plt.figure(1)
plt.subplot(311)
plt.plot(range(0, center_padded_trace.size), center_padded_trace, 'r-',
range(offset1, left_padded_trace.size+offset1), left_padded_trace, 'b--',
range(0, correlation1.size), correlation1/corr_max1, 'g-',
[corr_peak_index1], [1], 'k+')
plt.subplot(312)
plt.plot(range(0, center_padded_trace.size), center_padded_trace, 'r-',
range(offset2, center_padded_trace.size+offset2), center_padded_trace, 'b--',
range(0, correlation2.size), correlation2/corr_max2, 'g-',
[corr_peak_index2], [1], 'k+')
plt.subplot(313)
plt.plot(range(0, center_padded_trace.size), center_padded_trace, 'r-',
range(offset3, right_padded_trace.size+offset3), right_padded_trace, 'b--',
range(0, correlation3.size), correlation3/corr_max3, 'g-',
[corr_peak_index3], [1], 'k+')
plt.show()
x = np.arange(200)*0.01
y1 = np.array([*map(_main, x)])
y2 = np.array([*map(_main, x, np.ones(x.size,int))])
plt.figure(1)
plt.subplot(211)
plt.title('synthetic')
plt.plot(x,y1)
plt.legend(('left-shifted input', 'centered input', 'right-shifted input'))
plt.subplot(212)
plt.title('natural')
plt.plot(x,y2)
plt.ylabel('x-offset of result')
plt.xlabel('y-offset')
plt.savefig("summary.png")
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