Function iagaussian

Synopse

Generate a d-dimensional Gaussian image.

g = gaussian(s, mu, cov)
- g: Image.

s: Image shape. (rows columns)
mu: Image. Mean vector. n-D point. Point of maximum value.
cov: Covariance matrix (symmetric and square).



In [3]:

    
import numpy as np

def gaussian(s, mu, cov):
    d = len(s)  # dimension
    n = np.prod(s) # n. of samples (pixels)
    x = np.indices(s).reshape( (d, n))
    xc = x - mu 
    k = 1. * xc * np.dot(np.linalg.inv(cov), xc)
    k = np.sum(k,axis=0) #the sum is only applied to the rows
    g = (1./((2 * np.pi)**(d/2.) * np.sqrt(np.linalg.det(cov)))) * np.exp(-1./2 * k)
    return g.reshape(s)

Description

A n-dimensional Gaussian image is an image with a Gaussian distribution. It can be used to generate test patterns or Gaussian filters both for spatial and frequency domain. The integral of the gaussian function is 1.0.

Examples



In [1]:

    
testing = (__name__ == "__main__")
if testing:
    ! jupyter nbconvert --to python gaussian.ipynb
    import numpy as np
    import sys,os
    ia898path = os.path.abspath('../../')
    if ia898path not in sys.path:
        sys.path.append(ia898path)
    import ia898.src as ia
    
    %matplotlib inline
    import matplotlib.pyplot as plt









    



[NbConvertApp] Converting notebook gaussian.ipynb to python
[NbConvertApp] Writing 3260 bytes to gaussian.py

Example 1 - Numeric 2-dimensional



In [2]:

    
if testing:
    f = ia.gaussian((8, 4), np.transpose([[3, 1]]), [[1, 0], [0, 1]])
    print('f=\n', np.array2string(f, precision=4, suppress_small=1))
    g = ia.normalize(f, [0, 255]).astype(np.uint8)
    print('g=\n', g)









    



f=
 [[ 0.0011  0.0018  0.0011  0.0002]
 [ 0.0131  0.0215  0.0131  0.0029]
 [ 0.0585  0.0965  0.0585  0.0131]
 [ 0.0965  0.1592  0.0965  0.0215]
 [ 0.0585  0.0965  0.0585  0.0131]
 [ 0.0131  0.0215  0.0131  0.0029]
 [ 0.0011  0.0018  0.0011  0.0002]
 [ 0.      0.0001  0.      0.    ]]
g=
 [[  1   2   1   0]
 [ 20  34  20   4]
 [ 93 154  93  20]
 [154 255 154  34]
 [ 93 154  93  20]
 [ 20  34  20   4]
 [  1   2   1   0]
 [  0   0   0   0]]

Example 2 - one dimensional signal



In [3]:

    
# note that for 1-D case, the tuple has extra ,
# and the covariance matrix must be 2-D
if testing:
    f = ia.gaussian( (100,), 50, [[10*10]]) 
    g = ia.normalize(f, [0,1])
    plt.plot(g)
    plt.show()

Example 3 - two-dimensional image



In [4]:

    
if testing:
    f = ia.gaussian((150,250), np.transpose([[75,100]]), [[40*40,0],[0,30*30]])
    g = ia.normalize(f, [0,255]).astype(np.uint8)
    ia.adshow(g)

Example 4 - Numeric 3-dimensional



In [6]:

    
if testing:
    f = ia.gaussian((3,4,5), np.transpose([[1,2,3]]), [[1,0,0],[0,4,0],[0,0,9]])
    print('f=\n', np.array2string(f, precision=4, suppress_small=1))
    g = ia.normalize(f, [0,255]).astype(np.uint8)
    print('g=\n', g)









    



f=
 [[[ 0.0024  0.0031  0.0037  0.0039  0.0037]
  [ 0.0034  0.0045  0.0054  0.0057  0.0054]
  [ 0.0039  0.0051  0.0061  0.0064  0.0061]
  [ 0.0034  0.0045  0.0054  0.0057  0.0054]]

 [[ 0.0039  0.0051  0.0061  0.0064  0.0061]
  [ 0.0057  0.0075  0.0088  0.0093  0.0088]
  [ 0.0064  0.0085  0.01    0.0106  0.01  ]
  [ 0.0057  0.0075  0.0088  0.0093  0.0088]]

 [[ 0.0024  0.0031  0.0037  0.0039  0.0037]
  [ 0.0034  0.0045  0.0054  0.0057  0.0054]
  [ 0.0039  0.0051  0.0061  0.0064  0.0061]
  [ 0.0034  0.0045  0.0054  0.0057  0.0054]]]
g=
 [[[  0  23  40  47  40]
  [ 33  67  92 102  92]
  [ 47  86 115 125 115]
  [ 33  67  92 102  92]]

 [[ 47  86 115 125 115]
  [102 158 200 216 200]
  [125 189 237 254 237]
  [102 158 200 216 200]]

 [[  0  23  40  47  40]
  [ 33  67  92 102  92]
  [ 47  86 115 125 115]
  [ 33  67  92 102  92]]]

Equation

$$ f(x) = \frac{1}{\sqrt{2 \pi} \sigma} exp\left[ -\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2 \right] $$$$ f({\bf x}) = \frac{1}{(2 \pi)^{d/2}|\Sigma|^{1/2}} exp\left[ -\frac{1}{2}\left({\bf x} - \mu \right)^t\Sigma^{-1}\left({\bf x} - \mu \right)\right] $$