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Image reconstruction can help to determine how well an image may be characterized by a small finite set of its moments. Also, we can identify the number of descriptors needed to describe an image. In this work, we present a comparative analysis using different set of discrete orthogonal moments in terms of normalized image reconstruction error (NIRE). Color image reconstruction is performed with different color channels and various orders of different discrete orthogonal moments. Finally the results obtained by the reconstruction of three color images with different families of orthogonal moments and an error analysis to compare their capacity of description are presented, also the conclusions obtained from this work are presented.

Nowadays, all digital cameras acquire chromatic images, which are capable of recording complex details. The conventional approach to color image analysis consists of processing the red, green and blue channels separately using digital image processing techniques, and combines the individual output results [

In this article we present an analysis based on a reconstruction error of color images using different orders of moments, which allows knowing the capacity of reconstruction with different set of classical orthogonal moments is analyzed, also the recovery error graphs that were obtained are presented.

The discrete orthogonal moments have the ability to represent global features and describe its most important characteristics, which have a kernel base of discrete orthogonal polynomials [

The orthogonal moments are defined as follows:

φ m , n = ∑ x = 1 m ∑ y = 1 n f ( x , y ) P n ( x ) P m ( y ) (1)

where P n ( x ) and P m ( x ) are a set of discrete orthogonal f ( x , y ) is an image function and m, n are the image size.

Considering that a color image is formed by RGB channels they can be defined as,

f ( x , y , 1 ) = f R ( x , y ) , f ( x , y , 2 ) = f G ( x , y ) , f ( x , y , 3 ) = f B ( x , y ) , (2)

The calculation of the orthogonal moments is defined as

φ n , m c = ∑ x = 1 m ∑ y = 1 n f ( x , y , c ) P n ( x ) P m ( y ) (3)

where P n ( x ) y P m ( y ) are polynomial basic functions, f ( x , y , c ) is a color image, and c represents the RGB channel of the image.

Discrete Orthogonal PolynomialsA general way to obtain normalized discrete orthogonal polynomials p ˜ n ( x ) is by the following recurrence relation (H. Zhu et al. [

A p ˜ n ( x ) = B ∗ D p ˜ n − 1 ( x ) + C ∗ E p ˜ n − 2 ( x ) (4)

where A, B, C, D, E are terms independent of each of the polynomial sets shown in “

t ˜ n ( x ; N ) | k ˜ n ( x ; p , N ) | c ˜ n a 1 ( x ) | |
---|---|---|---|

A | n 2 ( 2 n − 1 ) | n | − a 1 |

B | x − N − 1 2 | x − n + 1 − p ( N − 2 n + 2 ) | x − n + 1 − a 1 |

C | − ( n − 1 ) ( N 2 − ( n − 1 ) 2 ) 2 ( 2 n − 1 ) | − p ( 1 − p ) ( N − n + 2 ) | n − 1 |

D | 2 n + 1 ( N 2 − n 2 ) ( n + 1 ) | n p ( 1 − p ) ( N − n + 1 ) | a 1 n |

E | 2 n + 1 ( N 2 − n 2 ) ( N 2 − ( n + 1 ) ) ∗ 1 2 n − 3 | n p ( 1 − p ) ( N − n + 2 ) ∗ n − 1 N − n + 1 | a 1 2 n ( n − 1 ) |

h ˜ n a , b ( x ; N ) | m ˜ n β , μ ( x ) | |
---|---|---|

A | n a + b + 2 n − 1 ∗ a + b + n a + b + 2 n | μ μ − 1 |

B | x − a − b + 2 N − 2 4 − ( b 2 + a 2 ) ( a + b + 2 N ) 4 ( a + b + 2 n − 2 ) ( a + b + 2 n ) | x − x μ − n + 1 − μ n + μ − β μ 1 − μ |

C | − ( a + n − 1 ) ( N − n + 2 ) a + b + 2 n − 1 ∗ ( a + b + N + n − 1 ) ( N − n + 1 ) a + b + 2 n − 1 | − ( n − 1 ) ( n − 2 + β ) 1 − μ |

D | n ( a + b + n ) ( a + b + 2 n + 1 ) ( a + n ) ( b + n ) ( a + b + n + N ) ( N − n ) ( a + b + 2 n − 1 ) | μ n ( β + n − 1 ) |

E | n ( n − 1 ) ( a + b + n ) ( a + n ) ( b + n ) ( a + n + 1 ) ( b + n + 1 ) ( N − n + 1 ) ( N − n ) ∗ ( a + b + n − 1 ) ( a + b + 2 n ) ( a + b + 2 n − 3 ) ( a + b + 2 n − 1 ) | μ 2 ( n − 1 ) ( β + n − 2 ) n ( β + n − 1 ) |

A special case of the discrete orthogonal polynomials are the Racah polynomials, which are calculated by the following relation of recurrence with respect to n [

A n U n α , β ( s , a , b ) = B n d n d n − 1 U n − 1 α , β ( s , a , b ) C n d n d n − 2 U n − 2 α , β ( s , a , b ) (5)

where A n , B n , C n are terms independent of Racah polynomials and U n − 1 α , β ( s , a , b ) ,

t ˜ n ( x ; N ) | k ˜ n ( x ; p , N ) | c ˜ n a 1 ( x ) | |
---|---|---|---|

p ˜ 0 ( x ) | 1 N | N ! p x ( 1 − p ) N − x x ! ( N − x ) ! | e − μ μ x x ! |

p ˜ 1 ( x ) | ( N − 1 − 2 x ) 3 N ( N 2 − 1 ) | − p ( N − x ) + x ( 1 − p ) ∗ ( N − 1 ) ! p x − 1 ( 1 − p ) N − x − 1 x ! ( N − x ) ! | μ − x μ e − μ μ x + 1 x ! |

h ˜ n a , b ( x ; N ) | m ˜ n β , μ ( x ) | |
---|---|---|

p ˜ 0 ( x ) | ( a + 1 ) b ( a + b + 1 ) ( N − a ) b + 1 | μ x ( β + x − 1 ) ! x ! ( β − 1 ) ! ( 1 − μ ) β |

p ˜ 1 ( x ) | ( a + b + 2 ) x − ( b + 1 ) ( N − 1 ) ∗ a + b + 3 ( a + 1 ) ( b + 1 ) ( n − 1 ) ( N + 1 + b + 1 ) | ( β + x − x μ ) μ x ( β + x − 1 ) ! x ! ( β − 1 ) ! μ ( 1 − μ ) β β |

U n − 2 α , β ( s , a , b ) are the initial values of Zero and first order polynomials shown in “

Below, the graphs obtained from the calculation of the first 5 polynomials of the six families of discrete orthogonal moments are shown in

One of the different applications of the moments’ functions with orthogonal base is the reconstruction of color images f ( x , y , c ) , which allows to determine the number of descriptors that can characterize the color image.

It is possible to reconstruct an image from its moments using the respective inverse transformation. The reconstruction of an image f ( x , y ) is given by [

f ˜ ( x , y ) = ∑ x = 1 m ∑ y = 1 n φ m , n P n ( x ) P m ( y ) , (6)

where P n ( x ) and P m ( x ) are polynomial basis functions and φ m , n are the moments of the image to be reconstructed.

The reconstruction of a color image f ( x , y , c ) is given by:

f ˜ c ( x , y ) = ∑ x = 1 m ∑ y = 1 n φ n , m c P n ( x ) P m ( y ) , (7)

where P n ( x ) y P m ( y ) are polynomial basis functions and φ n , m c are the moments of the image to be reconstructed in its three RGB channels.

p ˜ 0 ( x ) | ( 2 s + 1 ) ( p ( s ) ) d 0 2 |
---|---|

p ˜ 1 ( x ) | p 1 ( s ) − p ( s − 1 ) x ( s + 1 2 ) − x ( s − 1 2 ) 1 p ( s ) ∗ e − μ μ x + 1 x ! |

A n | n ( α + β + n ) ( α + β + 2 n − 1 ) ( α + β + 2 n ) |

B n | x − a 2 + b 2 + ( a − β ) 2 ( b + α ) 2 − 2 ( α + β + 2 n − 2 ) ( α + β − 2 ) 4 + ( α + β + 2 n − 2 ) − ( α + β + 2 n ) 8 − ( β 2 − α 2 ) [ ( b + α 2 ) 2 − ( a − β / 2 ) 2 ] 2 ( α + β + 2 n − 2 ) ( α + β + 2 n ) |

C n | − ( α + n − 1 ) ( β + n − 1 ) ( α + β + 2 n − 2 ) ( α + β + 2 n ) ∗ [ ( a + b + α − β 2 ) 2 ( n − 1 + α + β 2 ) 2 ] ∗ [ ( b + a + α + β 2 ) 2 ( n − 1 + α + β 2 ) 2 ] |

d n | Γ ( α + n + 1 ) Γ ( β + n + 1 ) Γ ( b − a + α + β + n + 1 ) Γ ( a + b + α + n + 1 ) ( α + β + 2 n + 1 ) n ! ( b − a − n − 1 ) Γ ( α + β + n + 1 ) Γ ( a + b − β − n ) |

ρ n ( s ) | Γ ( α + s + n + 1 ) Γ ( s − a + β + n + 1 ) Γ ( N + α − s ) Γ ( N + α + s + n + 1 ) Γ ( a − β + s + 1 ) Γ ( s − a + 1 ) Γ ( N − s − n ) Γ ( N + s + 1 ) |

To evaluate the efficiency of the moments, the reconstruction metric based on the normalized image reconstruction error (NIRE) is used, which is defined as the normalized square error between the input image f c ( x , y ) and the reconstructed image f ˜ c ( x , y , c ) , expressed by [

NIRE c = ∑ y = 1 N ∑ x = 1 M [ f c ( x , y ) − f ˜ c ( x , y ) ] 2 ∑ y = 1 N ∑ x = 1 M f c 2 ( x , y ) (8)

Therefore, the measurement for the reconstruction error of a color image that takes into account the three channels is given by:

MeanNIRE = NIRE R + NIRE G + NIRE B 3 (9)

Next are the results obtained with the proposed algorithm for video encryption and recovery of the following three grayscale test images with dimensions of 512 ×

512 pixels presented in

We have reconstructed these test images shown in

Moments | Reconstruction orders | |||
---|---|---|---|---|

10 | 25 | 50 | 100 | |

Tchebichef | ||||

Krawtchouk | ||||

Charlier | ||||

Hahn | ||||

Meixner | ||||

Racah |

Moments | Reconstruction orders | |||
---|---|---|---|---|

10 | 25 | 50 | 100 | |

Tchebichef | ||||

Krawtchouk | ||||

Charlier | ||||

Hahn | ||||

Meixner | ||||

Racah |

The NIRE graphics of the reconstruction comparison of the three images with the moments sets are shown in

According to the value distributions of each set of orthogonal polynomials as shown in

Moments | reconstruction orders | |||
---|---|---|---|---|

10 | 25 | 50 | 100 | |

Tchebichef | ||||

Krawtchouk | ||||

Charlier | ||||

Hahn | ||||

Meixner | ||||

Racah |

expected with these polynomials.

Rectangular moments were used for the reconstruction of an image, with which it was possible to examine the performance of the different moments used for the representation of the information, as well as its convergence through the number of moments necessary for the reconstruction of the images.

As can be seen in the results obtained shown in Tables 6-8. The Tchebichef, Hahn and Racah polynomials present a better capacity of description, since they allow to recover the image with a smaller number of reconstruction orders. In

this case with more than 50 orders it can recover almost the entire image.

J. S. Rivera-López thanks CONACyT for the scholarship with number 423649.

Rivera-López, J.S. and Camacho-Bello, C.J. (2017) Color Image Reconstruction by Discrete Orthogonal Moment. Journal of Data Analysis and Information Processing, 5, 156-166. https://doi.org/10.4236/jdaip.2017.54012