In [1]:

Copied!

#%matplotlib widget
#%matplotlib widget

Object moments¶

We have seen that moments can be used to describe contours, similarily, the content of the shape can be described by this feature.

definition The definition in the continuous domain is as follow

$$ m_{pq} = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} x^p y^q f(x,y)\:dx\:dy $$$$ \mu_{pq} = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} (x-\bar x)^p (y-\bar y)^q f(x,y)\:dx\:dy $$$$ \bar{x}=\frac{m_{10}}{m_{00}},\bar{y}=\frac{m_{01}}{m_{00}} $$

for a discrete domain, which is the case for image of pixels, we have

$$ m_{pq} = \sum_{x= -\infty}^{+\infty}\sum_{y=-\infty}^{+\infty} x^p \; y^q \; f(x,y) $$

to eliminate the influence of the absolute shape position, one define the centered moments $\mu$

$$ \mu_{pq} = \sum_{x= -\infty}^{+\infty}\sum_{y=-\infty}^{+\infty} (x-\bar x)^p \; (y-\bar y)^q \; f(x,y) $$

to eliminate the influence of the scale, one define $\eta$ as the normalized $\mu$

$$ \eta_{pq} = \frac{\mu_{pq}}{\mu_{00}^\gamma} $$

with

$$ \gamma = \frac{p+q}{2} + 1 $$

Up to here, $\eta_{pq}$ are independent to translation and scale, by are still dependent to the rotation

Hu defined invariants, called Hu's invariants, that combined $\eta_{pq}$ having the rotation invariance property,

$$ \begin{align} I_1 =\ & \eta_{20} + \eta_{02} \\ I_2 =\ & (\eta_{20} - \eta_{02})^2 + (2\eta_{11})^2 \\ I_3 =\ & (\eta_{30} - 3\eta_{12})^2 + (3\eta_{21} - \eta_{03})^2 \\ I_4 =\ & (\eta_{30} + \eta_{12})^2 + (\eta_{21} + \eta_{03})^2 \\ I_5 =\ & (\eta_{30} - 3\eta_{12}) (\eta_{30} + \eta_{12})[ (\eta_{30} + \eta_{12})^2 - 3 (\eta_{21} + \eta_{03})^2] + \\ \ & (3\eta_{21} - \eta_{03}) (\eta_{21} + \eta_{03})[ 3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2] \\ I_6 =\ & (\eta_{20} - \eta_{02})[(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2] + 4\eta_{11}(\eta_{30} + \eta_{12})(\eta_{21} + \eta_{03}) \\ I_7 =\ & (3\eta_{21} - \eta_{03})(\eta_{30} + \eta_{12})[(\eta_{30} + \eta_{12})^2 - 3(\eta_{21} + \eta_{03})^2] + \\ \ & (\eta_{30} - 3\eta_{12})(\eta_{21} + \eta_{03})[3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2]. \end{align} $$

High order invariants can be very sensitive to noise, due to the high order exponent in the sum.

see also Moments p514 Digital image processing.Gonzalez, Rafael C. 2009. Pearson education.

In [2]:

Copied!





import numpy as np
from skimage.morphology import disk, square
import matplotlib.pyplot as plt
from skimage.measure import label,regionprops
from skimage.io import imread
from scipy.spatial.distance import pdist,squareform

ima = imread('../../data/alphabet.png')[:,:,0]==0
lab,nlab = label(ima,return_num=True)

# extract label features
props = regionprops(lab)

hu = []
for p in props:
    hu.append(p.moments_hu)
hu = np.asarray(hu)

# normalize Hu's features
mean_hu = np.mean(hu,axis=0)
std_hu = np.std(hu,axis=0)
norm_hu = (hu-mean_hu)/std_hu

# compute pairwize distance based on normalized Hu's features
dist = squareform(pdist(norm_hu))

plt.figure(figsize=[10,10])
plt.imshow(dist,interpolation='nearest');
plt.show()
import numpy as np
from skimage.morphology import disk, square
import matplotlib.pyplot as plt
from skimage.measure import label,regionprops
from skimage.io import imread
from scipy.spatial.distance import pdist,squareform

ima = imread('../../data/alphabet.png')[:,:,0]==0
lab,nlab = label(ima,return_num=True)

# extract label features
props = regionprops(lab)

hu = []
for p in props:
    hu.append(p.moments_hu)
hu = np.asarray(hu)

# normalize Hu's features
mean_hu = np.mean(hu,axis=0)
std_hu = np.std(hu,axis=0)
norm_hu = (hu-mean_hu)/std_hu

# compute pairwize distance based on normalized Hu's features
dist = squareform(pdist(norm_hu))

plt.figure(figsize=[10,10])
plt.imshow(dist,interpolation='nearest');
plt.show()

searching for nearest resembling object, using Hu's invariant features

In [3]:

Copied!





idx = np.argsort(dist[1,:])
plt.figure()
for j,i in enumerate(idx[:16]):
    bb = props[i].bbox
    crop = ima[bb[0]:bb[2],bb[1]:bb[3]]
    plt.subplot(4,4,j+1)
    plt.imshow(crop)
plt.show()
idx = np.argsort(dist[1,:])
plt.figure()
for j,i in enumerate(idx[:16]):
    bb = props[i].bbox
    crop = ima[bb[0]:bb[2],bb[1]:bb[3]]
    plt.subplot(4,4,j+1)
    plt.imshow(crop)
plt.show()

In [ ]: