The authors offer a method for detecting the edges of texture objects in remote sensing images. This method is based on the evaluation of textural and brightness attributes. It is proposed to use transition probabilities for three-dimensional Markov chains with two states as texture features, averaged within a sliding window. It makes possible to improve the detection accuracy of texture objects on multichannel or multi-time snapshots. To reduce the computational resources, it is proposed to determine the signs by the bit planes of the senior, most informative digits of the digital image. The simulation results confirm the effectiveness of the proposed method.
One of the new approaches in the selection of textures is the joint use of spectral and textural
features [
Many of these methods require large computational resources and do not accurately separate the attribute space. The quality of the selection of texture objects depends on the choice of the most appropriate characteristics of the task and approaches to the development of algorithms. In addition, as multispectral images have a high spatial resolution, an important factor in the detection of textures is to reduce the computational complexity of algorithms. In this regard, the use of known methods is not always advisable.
As the texture regions on images occupy extended space with homogeneous statistical
characteristics and different for different regions, the method based on the mathematical apparatus of
Markov chains [
Works [
When using the mathematical apparatus of Markov chains, textural features are probabilistic
characteristics between elements of images. Works [
Considering that there is a large statistical dependence between separate areas of images taken in different spectral ranges (channels), it is proposed to use the nature of the statistical connection not only between the elements inside the DI, but also the inter-channel DI. The use of the transition probability for three-dimensional Markov chains as a textural attribute improves the accuracy of prediction of image elements and, accordingly, the quality of the selection of textures. Also, an approach based on three-dimensional Markov chains can be applied to detect changed texture regions in time-varying DIs.
Real remote sensing DIs contain objects in which the pixel brightness changes slightly (water bodies, fields of different crops, etc.). Therefore, to improve the quality of detection of texture objects for which the probabilistic characteristics between the elements are close to unity, it is advisable to take into account the brightness characteristics determined by BBI of the senior bits DI.
The aim of the article is to develop a method of segmentation of multi-channel digital images with high spatial resolution based on the mathematical apparatus of three-dimensional Markov chains, which allows to increase the accuracy of detecting the edges of texture objects while reducing computational resources.
A special case of multichannel images can be considered color RGB images. Therefore, the proposed method will be considered for the case of three-component DI with R, G, and B color channels. Each RGB color component of an image is a g-digit digital halftone image. There is a significant statistical relationship between the elements of individual DI areas belonging to different color channels. For example, objects of yellow color are well expressed on green and red components, and objects of white color are seen on all three components. Therefore, taking into account the statistical connection between the elements inside DI and between individual color channels, we can assume that RGB images allow them to be approximated by a three-dimensional Markov chain with several states, and BBI with a three-dimensional Markov chain with two states and horizontal transition matrixes 1Π = 1π i(jl) 2×2 , vertical 2 Π = 2π i(jl) 2×2 and 4 Π = 4π i(jl) 2×2 between channels. The introduction of the inter-channel matrix 4 Π = 4π i(jl) 2×2 possible to get rid of the segmentation of the separate components and identify the changed areas in the images.
Figure 1 shows a fragment of a three-dimensional binary Markov field corresponding to a fragment of an BBI of a two-channel DI. will make it ν(l ) 1 ν(l )
2 ν(l ) 4 ν(l ) 3
The amount of information in the element ν 4(l ) relative to the elements of the nearest neighborhood Λi, j,k ={ν1(l) , ν(2l) , ν(l)} , in accordance with the mathematical model of the three-dimensional random 3
I (ν 4(l) v1(l) , v2(l) ,ν 3(l) ) = − log i=1
Markov process presented in [
3 ∏ w(ν 4(l) ν i(l) )w(ν 4(l) v1(l) , v2(l) ,ν 3(l) ) ∏ w(ν 4(l) ν i(l) ,ν (jl) ) where w(ν 4(l) ) is the probability density of transitions in Markov chains of various dimensions; the products in (1) are calculated for all possible non-coinciding combinations of various subscript indices.
The probability density of transitions in a three-dimensional binary Markov chain w(ν 4(l ) | Λi(,l )j.k ) can have the form:
2 w(ν 4(l ) | Λi(,l )j,k ) =∑ π (ν 4(l ) i,...,r=1 =M i(l ) |ν 1(l ) =M (jl ) ;ν 2(l ) =M k(l ) ; ν (l) = M r(l ) ) ×δ (ν 1(l ) − M (jl ) )δ (ν 2(l ) − M k(l ) )δ (ν 3(l) − M r(l ) ).
3 where δ (⋅) is the delta function.
The probabilities of the statesπ i(ilii) of an elementν 4(l ) are determined by the argument of expression (1) and for various combinations of neighboring elements Λ(l) i, j,k can be calculated using the formulae presented in Table 1, where rπ (l) ( r = 1, 7) are elements of transition probability matrices in oneii dimensional Markov chains with two states of three main - 1Π(l ) , 2 Π(l ) , 4 Π(l ) and four related matrices 3 Π(l ) =1Π(l ) × 2 Π(l ) ; 5 Π(l ) =1Π(l ) ⋅ 4 Π(l ) ; 6 Π(l ) =2Π(l ) ⋅ 4 Π(l ) ; 7 Π(l ) =3Π(l ) ⋅ 4 Π(l ). (1) (2) 1π i(il) ⋅ 2π i(il) ⋅ 4π i(jl) ⋅ 7π i(jl) ,
3π i(il) ⋅ 5π i(jl) ⋅ 6π i(jl) 1π i(jl) ⋅ 2π i(jl) ⋅ 4π i(jl) ⋅ 7π i(jl) ,
3π i(il ) ⋅ 5π i(il ) ⋅ 6π i(il ) 1π i(jl) ⋅ 2π i(jl) ⋅ 4π i(jl) ⋅ 7π i(jl) ,
3π i(il ) ⋅ 5π i(il ) ⋅ 6π i(il ) 1π i(jl) ⋅ 2π i(jl) ⋅ 4π i(il) ⋅ 7π i(il) ,
3π i(il ) ⋅ 5π i(jl ) ⋅ 6π i(jl ) 1π i(il) ⋅ 2π i(il) ⋅ 4π i(il) ⋅ 7π i(il) , 3π i(il ) ⋅ 5π i(il ) ⋅ 6π i(il ) (3) (l) estimation πˆiiii using formulas (3), as well as estimating brightness in BBI of the senior, most informative DI digits.
Taking into account the local changes in the probability and brightness characteristics on multichannel images, a three-dimensional sliding window was used for calculation.
The texture features were the average estimates corresponding to the central element of the window - πi(ilii,r,k ) of probability of transitions in the three-dimensional Markov chain and the brightness L(l,r,k ) of the BBI: πi(ilii,r,k ) = m1× n r∑m=1 k∑n=1πˆi(ilii,r,k ) . (4) L(l,r,k ) = 1 ∑m ∑n Lˆ(l,r,k ) . (5)
m × n r =1 k =1
The pixel belonging to one or another texture object in the image of the k- channel was carried out on the basis of the analysis of the texture feature histogram. The number of peaks in the histogram corresponded to the number of texture objects with different probabilistic characteristics. The threshold value was chosen as the minimum value between two adjacent peaks of the histogram. When evaluation πi(ilii,r,k ) was close to 1, the averaged brightness was additionally used to decide whether a pixel belongs to one or another object. Each texture object was assigned its own label.
The authors combined images of different channels in order to increase the information content of texture objects.
Artificial and real DI were used for experimental studies. Artificial binary images were generated by a
given markup using a mathematical model based on a two-dimensional Markov chain and the
algorithm given in [
To assess the accuracy of the selection of the edges of texture objects, the authors compared the segmented image with the ideal markup and then calculated the number of erroneously segmented elements:
ESE = 1 ∑h ∑w F (i, j) (6)
h ⋅ w i=1 j=1 where h, w are image height and width; F is a value that is equal to zero when the image element is segmented correctly, and equal to 1 if otherwise.
Examples of the results of segmentation of artificial images are shown in Figure 2: (a) an image containing two areas with different statistical characteristics; (b) the result of a segmentation algorithm based on a three-dimensional Markov chain with a difference between the probabilities ∆ of segments in 0.25; (в) an image with three objects; (d) the result of a segmentation algorithm based on an estimation of probability πi(ilii,r,k ) and brightness .
а)
b) c) Figure 2. Results of the segmentation of artificial images. d) 0.8 0.9 6.31 5.88
Three-dimensional segmentation for all values of the transition probabilities gave a segmentation error less than segmentation based on two-dimensional Markov chains. The worst segmentation result was obtained when the difference between the probabilities of the segments was 0.1. As this difference increases, the segmentation error is a fraction of a percent.
Table 3 presents the estimates of the ESE criterion for segmented binary images, similar to Fig. 2c. Three-dimensional segmentation was performed on the basis of estimates of probability and brightness characteristics within a sliding window of 11x11.
area 1 = 2
Accounting for the brightness of the image in a given example allowed to reduce the segmentation error by up to 18 times.
An example of real image remote sensing segmentation, containing four types of objects (urban buildings, forest and fields planted with different crops) is shown in Figure 3: (a) - RGB image; (b) and (c) - BBI of the 7th category of channels R and B, which were used to select the edges of texture objects; (d) and (d) are the results of two-dimensional segmentation of the image of the R and B channels, respectively; (e) - the result of three-dimensional segmentation of channel R relative to channel B.
Two-dimensional segmentation in the images of the R and B channels allowed to detect only 3 objects. The introduction of the matrix of probabilities of transitions between color components 4 Π = 4π i(jl) 2×2 allowed to get rid of the segmentation of the separate components and to detect another object. The proportion of erroneously segmented elements is 7.8%. d)
Figure 4. The result of detecting the edges of the modified texture objects in multi-time shots.
The developed segmentation method, based on an estimate of transition probabilities for
threedimensional Markov chains and luminosity, improved the accuracy of detecting the edges of extended
texture objects on multichannel and time-varying images compared to the two-dimensional
segmentation algorithm proposed in [