python - 使用python对图像使用最大似然算法进行分割

Question

我想使用在 python 中实现的最大似然算法来执行图像分割。类别的平均向量和协方差矩阵是已知的，并且迭代图像(相当大...5100X7020)我们可以计算每个像素属于给定类别的概率。

简单地用 Python 编写

import numpy as np
from numpy.linalg import inv
from numpy.linalg import det
...

probImage1 = []
probImage1Vector = []

norm = 1.0 / (np.power((2*np.pi), i/2) * np.sqrt(np.linalg.det(covMatrixClass1)))
covMatrixInverz = np.linalg.inv(covMatrixClass1)
for x in xrange(x_img):
    for y in xrange(y_img):
        X = realImage[x,y]
        pixelValueDifference = X - meanVectorClass1
        mult1 = np.multiply(-0.5,np.transpose(pixelValueDifference))
        mult2 = np.dot(covMatrixInverz,pixelValueDifference)
        multMult = np.dot(mult1,mult2)
        expo = np.exp(multMult)     
        probImage1Vector.append(np.multiply(norm,expo))
    probImage1.append(probImage1Vector)
    probImage1Vector = []

此代码在大图像上执行时非常慢的问题...像向量减法和乘法这样的计算会消耗大量时间，即使它们只是 1X3 向量。

能否请您提示如何加快这段代码的速度？我真的很感激。抱歉，如果我不清楚，我仍然是 python 的初学者。

Answer 1

仔细看看:

mult1 = np.multiply(-0.5,np.transpose(pixelValueDifference))
mult2 = np.dot(covMatrixInverz,pixelValueDifference)
multMult = np.dot(mult1,mult2)

我们看到操作基本上是:

A.T (d) C (d) A         # where `(d)` is the dot-product

这三个步骤可以很容易地表示为 np.einsum 中的一个字符串符号, 像这样 -

np.einsum('k,lk,l->',pA,covMatrixInverz,-0.5*pA)

在迭代器 i(=x) 和 j(=y) 上执行此操作，我们将得到一个完全向量化的表达式 -

np.einsum('ijk,lk,ijl->ij',pA,covMatrixInverz,-0.5*pA))

或者，我们可以使用 np.tensordot 执行减和的第一部分 -

mult2_vectorized = np.tensordot(pA, covMatrixInverz, axes=([2],[1]))
output = np.einsum('ijk,ijk->ij',-0.5*pA, mult2_vectorized)

---

基准测试

将所有方法列为函数 -

# Original code posted by OP to return array
def org_app(meanVectorClass1, realImage, covMatrixInverz, norm):
    probImage1 = []
    probImage1Vector = []
    x_img, y_img = realImage.shape[:2]
    for x in xrange(x_img):
        for y in xrange(y_img):
            X = realImage[x,y]
            pixelValueDifference = X - meanVectorClass1
            mult1 = np.multiply(-0.5,np.transpose(pixelValueDifference))
            mult2 = np.dot(covMatrixInverz,pixelValueDifference)
            multMult = np.dot(mult1,mult2)
            expo = np.exp(multMult)     
            probImage1Vector.append(np.multiply(norm,expo))
            probImage1.append(probImage1Vector)
            probImage1Vector = []
    return np.asarray(probImage1).reshape(x_img,y_img)
    
def vectorized(meanVectorClass1, realImage, covMatrixInverz, norm):
    pA = realImage - meanVectorClass1
    mult2_vectorized = np.tensordot(pA, covMatrixInverz, axes=([2],[1]))
    return np.exp(np.einsum('ijk,ijk->ij',-0.5*pA, mult2_vectorized))*norm
    
def vectorized2(meanVectorClass1, realImage, covMatrixInverz, norm):
    pA = realImage - meanVectorClass1
    return np.exp(np.einsum('ijk,lk,ijl->ij',pA,covMatrixInverz,-0.5*pA))*norm

时间 -

In [19]: # Setup inputs
    ...: meanVectorClass1 = np.array([23.96000000, 58.159999, 61.5399])
    ...: 
    ...: covMatrixClass1 = np.array([[ 514.20040404,  461.68323232,  364.35515152],
    ...:        [ 461.68323232,  519.63070707,  446.48848485],
    ...:        [ 364.35515152,  446.48848485,  476.37212121]])
    ...: covMatrixInverz = np.linalg.inv(covMatrixClass1)
    ...: 
    ...: norm = 0.234 # Random float number
    ...: realImage = np.random.rand(1000,2000,3)
    ...: 

In [20]: out1 = org_app(meanVectorClass1, realImage, covMatrixInverz, norm )
    ...: out2 = vectorized(meanVectorClass1, realImage, covMatrixInverz, norm )
    ...: out3 = vectorized2(meanVectorClass1, realImage, covMatrixInverz, norm )
    ...: print np.allclose(out1, out2)
    ...: print np.allclose(out1, out3)
    ...: 
True
True

In [21]: %timeit org_app(meanVectorClass1, realImage, covMatrixInverz, norm )
1 loops, best of 3: 27.8 s per loop

In [22]: %timeit vectorized(meanVectorClass1, realImage, covMatrixInverz, norm )
1 loops, best of 3: 182 ms per loop

In [23]: %timeit vectorized2(meanVectorClass1, realImage, covMatrixInverz, norm )
1 loops, best of 3: 275 ms per loop

看起来完全矢量化的 einsum + tensordot 混合解决方案做得很好!

为了进一步提升性能，还可以查看 numexpr module加速大型数组的指数计算。