Function h2stats

Synopse

The h2stats function computes several statistics given an image histogram.

g = h2stats(h)
- Output
  - g: unidimensional array. Array containing the statistics from the histogram
- Input
  - h: 1-D ndarray: histogram

Description

The h2stats function extracts some relevant statistics of the images where the histogram was computed:

[0] Mean (mean grayscale value)
[1] Variance (variance of grayscale values)
[2] Skewness
[3] Kurtosis
[4] entropy
[5] mode (gray scale value with largest occurrence)
[6] Percentile 1%
[7] Percentile 10%
[8] Percentile 50% (This is the median gray scale value)
[9] Percentile 90%
[10] Percentile 99%

Function Code



In [1]:

    
def h2stats(h):
    import numpy as np
    import ia898.src as ia

    hn = 1.0*h/h.sum() # compute the normalized image histogram
    v = np.zeros(11) # number of statistics

    # compute statistics
    n = len(h) # number of gray values
    v[0]  = np.sum((np.arange(n)*hn)) # mean
    v[1]  = np.sum(np.power((np.arange(n)-v[0]),2)*hn) # variance
    v[2]  = np.sum(np.power((np.arange(n)-v[0]),3)*hn)/(np.power(v[1],1.5))# skewness
    v[3]  = np.sum(np.power((np.arange(n)-v[0]),4)*hn)/(np.power(v[1],2))-3# kurtosis
    v[4]  = -(hn[hn>0]*np.log(hn[hn>0])).sum() # entropy
    v[5]  = np.argmax(h) # mode
    v[6:]  = ia.h2percentile(h,np.array([1,10,50,90,99])) # 1,10,50,90,99% percentile
    return v

Examples



In [1]:

    
testing = (__name__ == "__main__")

if testing:
    ! jupyter nbconvert --to python h2stats.ipynb
    import numpy as np
    import sys,os
    import matplotlib.image as mpimg
    ia898path = os.path.abspath('../../')
    if ia898path not in sys.path:
        sys.path.append(ia898path)
    import ia898.src as ia









    



[NbConvertApp] Converting notebook h2stats.ipynb to python
[NbConvertApp] Writing 4683 bytes to h2stats.py

Numeric Example



In [2]:

    
if testing:
    f = np.array([1,1,1,0,1,2,2,2,1])
    h = ia.histogram(f)
    print('statistics =', ia.h2stats(h))









    



statistics = [ 1.22222222  0.39506173 -0.20992233 -0.62109375  0.93688831  1.          0.08
  0.8         1.          2.          2.        ]

Image Example



In [3]:

    
if testing:
    %matplotlib inline
    import matplotlib.pyplot as plt
    import matplotlib.image as mpimg
    from scipy.stats import mode, kurtosis, skew, entropy
    
    f = mpimg.imread('../data/cameraman.tif')  
    plt.imshow(f,cmap='gray')
    h = ia.histogram(f)
    v = ia.h2stats(h) 
    print('mean =',v[0])
    print('variance =',v[1])
    print('skewness =',v[2])
    print('kurtosis = ',v[3])
    print('entropy = ',v[4])
    print('mode = ',v[5])
    print('percentil 1% = ',v[6])
    print('percentil 10% = ',v[7])
    print('percentil 50% = ',v[8])
    print('percentil 90% = ',v[9])
    print('percentil 99% = ',v[10])









    



mean = 137.065933228
variance = 7604.25522007
skewness = -0.554976386125
kurtosis =  -1.4152064701
entropy =  4.99579946489
mode =  14.0
percentil 1% =  2.0
percentil 10% =  8.0
percentil 50% =  180.0
percentil 90% =  222.0
percentil 99% =  251.0

Equation

The follow equations represent the statistics equations, where N denotes the number of intensity levels in the imagem ( usually between 0 and 255) and $p(x)$ is the probability density function, i.e. the normalized histogram (h) of the image, and X is a random variable that represents the pixels intensities $x_i$.

Mean

$$ \mu (X) = \sum_{i=1}^{N} x_i p(x_i) $$

Variance

$$ Var (X) = \sum_{i=1}^{N} (x_i - \mu)^2 p(x_i) $$

Skewness

$$ Skewness = \frac{\sum_{i=1}^{N} (x_i - \mu)^3 p(x_i)}{Var^{1.5}} $$

Kurtosis

$$ Kurtosis = \frac{\sum_{i=1}^{N} (x_i - \mu)^4 p(x_i)}{Var^{2}} -3 $$

Entropy

$$ Entropy(X) = - \sum_{i=1}^{N} p(x_i) log_{2}(p(x_i)) $$

Mode

The value that appears most often.

Median

It is the numerical value separating the higher half of a data sample or a probability distribution.

Percentil

O k-ésimo percentil Pk é o valor xk que corresponde à frequência cumulativa de N k/100, onde N é o tamanho amostral.

Dessa forma:

O 1º percentil determina o 1 % menor dos dados
O 98º percentil determina o 98 % menor dos dados
O 50º percentil determina o 50 % menor dos dados, representa a mediana

A definição para o p-ésimo percentil de N valores ordenados é correspondente ao valor que ocupa a posição k dada abaixo:

$$ k = \frac{p(n+1)}{100} $$

References

Contributions

Mariana Bento, August 2013



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