Abstract: - A simple algorithm for measuring the similarity between multi-column histograms is presented. The

proposed algorithm is intended for texture segmentation of images using histograms as texture features. The

purpose of developing such a specialized algorithm is to more accurately determine the boundaries between

neighboring texture segments. The algorithm is specially designed so that to express the similarity value as a

percentage. The main peculiarity of the proposed algorithm is that when calculating the similarity value, it

considers not only the corresponding histogram columns but also takes into account their neighboring

components. Due to this, the algorithm more adequately evaluates the similarity of histograms. The proposed

algorithm was implemented as a computer program as an integral part of the image segmentation model. The

efficiency of the histogram comparison algorithm is indirectly confirmed by the texture segmentation results of

This paper considers the problem of evaluating the

similarity between histograms, which are used as

texture features for the task of dividing an image

into texture segments. The problem of texture

segmentation of images is a key one for the analysis

of natural visual scenes of various natures:

landscapes, satellite photographs, and medical

images used by medical professionals in diagnosing

diseases.

need to be extracted from the image, are provided.

Using this information, the parameters of the

segmentation algorithm can be appropriately tuned,

for example, through training. This approach

vectors of large dimensions. Distance is a measure

of the "dissimilarity" or "difference" of vectors - for

example, the Euclidean distance, Manhattan

distance, Hamming distance, Minkowski distance,

Mahalanobis distance, Bulldozer's distance (earth

mover's distance, EMD (see exhaustive review,

[29]).

to use in the texture segmentation task. That's why

we have tried to construct an efficient algorithm for

histograms but also the components of their close

surroundings.

To evaluate the texture characteristics of different

areas of the image, texture windows of the same size

are used that cover the entire image (with overlaps).

The texture characteristics measured in the texture

windows serve to assess the similarity/difference

white images, the brightness range is 0 – 255. So,

the brightness histogram consists of 256 columns,

according to the brightness range of the pixels in the

image. The height of each histogram column

represents the number of pixels in the texture

window that have corresponding brightness values.

The maximum height of the columns is equal to the

number of pixels in the texture window. The texture

windows of 15×15 pixels were used in the

experiments. Therefore, the maximum height of the

the values of the histogram components are

interconnected in such a way that the sum of all

components of the histogram is constant. This

interdependence of the histogram components

makes it possible to somewhat improve the method

corresponding columns Pi and Di – by Δi. Δi is equal

to the smallest height of the two columns Pi and Di,

i.e., Δi = min (Pi, Di). Alternatively, Δi can also be

represented by the formula

Δi = (Pi + Di – |Pi – Di|) / 2, (1)

where i = 0, 1, 2, ..., I.

into account not only the corresponding components

of both histograms but also the components of their

close surroundings.

comparisons of each i-th component of the

histogram P - column Pi, with (1 + 2M) components

of the histogram D, where M is the radius of the

close surroundings of each i-th component of the

histogram P. Let us denote the similarity of the

histograms P and D by R(M), at the beginning, R(M) =

component is compared with the Dj component,

where j = i+(–)m, m = 0, 1, 2, 3, …, M.

height Dj is reduced by multiplying it by some

(between the Pi component and the Dj component) is

used in the process of calculating the similarity

between histograms R(M). That is, the more m=|i – j|,

the less impact the Dj component should have on the

value R(M). For this purpose, the coefficients Km < 1

are introduced, which reduce the proportion of the

size of the intersection Δ between the compared

columns of both histograms.

KM correlate with each other as follows: K0 > K1 >

(1 + 2M) values of Δm are calculated. Among these

(1 + 2M) values of Δm, the maximum value is

chosen, denoted by Δimax:

1+2M

Δimax = MAX (Δm) (4)

m = 0

value of histograms R(M). If, however, Pi > Δimax,

to zero, accordingly, S = 0%. Thus, the range of S is

0% – 100%.

windows differ significantly from each other,

mainly in that the non-zero components of both

histograms are located at mismatched positions

within the histograms. From the point of view of

common sense and a functional point of view (for

solving the problem of texture segmentation), these

histograms should be qualified as completely

different. And, accordingly, the algorithm proposed

in this paper specially designed for efficient

separation of texture segments, estimates the

the non-zero components of both histograms occupy

approximately the same position within the

histograms, and the corresponding columns have

significant intersections.

histograms calculated by Eq, (3) is 48.0, and the

percentage of similarity, converted from this

number, is 85%. The similarity value of the

histograms shown in Figure 2 (Appendix),

calculated by pair-wise component comparison

К10 were calculated by the formula Кm = (1 – 0.1 m).

The use of such coefficients led to the following

histograms produced by the proposed algorithm is

77%.

brightness histograms is the degree of

proposed algorithm provides a 100% diapason of

percentage similarity between the histograms being

compared (from complete similarity to complete

difference). This is the main advantage of the

algorithm and its contribution/novelty that makes it

possible to attain a more accurate determination of

the boundaries between the texture segments present

in the analyzed image.

the software implementation of the proposed

algorithm, only addition, subtraction, and

multiplication operations are used, due to that the

algorithm is fast and computationally effective.

segmentation of images into homogeneous texture

regions is confirmed by the segmentation results in

experiments on processing different natural images.

The results obtained in the experiments demonstrate

the effectiveness of the segmentation algorithm and

show that this algorithm performs correct (from a

human point of view) texture segmentation of a

wide range of images, [19], [20], [21], [22]. Thus,

the effectiveness of the key operation of the

segmentation algorithm, the histogram comparison

the segmentation algorithm are shown in different

colors. Areas of the image containing small texture

areas and borders between texture segments are

marked in white.

research is connections with the approaches

developed in the framework of Hyper Dimensional

Computing, [24], [25], [26], [27], [28].

[11] Fauzi, M.F., & Lewis, P.H. (2003). A Fully

[12] P.F. Felzenszwalb, D.P. Huttenlocher,

[14] H. Wei, M. Bartels, Unsupervised

[15] L. Wolf, X. Huang, I. Martin, D. Metaxas,

[18] J. Melendez, D. Puig, M.A. Garcia, Multi-

[19] A. Goltsev, V. Gritsenko, Algorithm of

APPENDIX