The formation of operators with given properties to solve original image processing tasksPattern Recognition and Image Analysis


A. I. Novikov
Computer Vision and Pattern Recognition / Computer Graphics and Computer-Aided Design


ISSN 10546618, Pattern Recognition and Image Analysis, 2015, Vol. 25, No. 2, pp. 230–236. © Pleiades Publishing, Ltd., 2015. 1 This paper is an extended version of the report sub mitted by the sponsor of the 11th International Con ference “Pattern Recognition and Image Analysis:

New Information Technologies (PRIA112013),”

Samara, Russia, Sept. 23–28, 2013 [1]. The transfor mation of an original image at each of its points with the help of operators with finite memory (with final sizes of sliding window) is often used for smoothing images and detecting brightnesschange boundaries in images (let’s call these procedures primary image pro cessing) received from different spectral sensors (tele vision, thermal, etc.). The set of weight factors (mask, pattern) of such an operator in its simplest case (under line by line or column processing) is the vector (1) in its common case a square matrix A = (αij) of the order m (sometimes matrix A is selected to be rectan gular). The factors α in (1) meet to one of the normal izing conditions = 1 or = 0.

Let fij, i = , j = be the values of brightness function f(x, y) on the grid having the size M × N. Usu ally the length of the sliding window m is an odd num ber m = 2k + 1 and the components of vector located at the same distance as the central element of the α0 are equal, i.e., 1 This paper uses the materials of the report submitted at the 11th

International Conference “Pattern Recognition and Image

Analysis: New Information Technologies,” Samara, Russia,

September 23–28, 2013. a α1 α2 … αm, , ,( ),= αjj 1= m ∑ αjj 1= m ∑ 1 M, 1 N, a a α k– α k– 1+ … α 1– α0 α1 … αk 1– αk, , , , , , , ,( ),=

The estimation of the brightness function in this case (for linebyline processing) is the formula

It is known that the components of the vector can be selected in such a way that the best estimation of brightness function in the specific sense will be found for an arbitrary point of sliding window [2] (2)

We introduce the operator Am, λ proving the scalar product ( , ) in (2). Index m in the designation of the operator determines the length of the sliding win dow (size of mask, pattern). Index λ determines the amount of left shift with regard to the point j0 when computing the scalar product ( , ).

Operator Am, λ, depending on factors set α, can pro vide both image smoothing and acquisition on first order partialderivative rates (gradient methods) or secondorder partialderivative rates (methods on sec ond order).

For example, 1) operator A7, 3 with weight factors = (–2, 3, 6, 7, 6, 3,–2) will provide original data smoothing with smoothing factor simultaneously providing an unbiased estimation of polynomials up to the third degree, inclusive; α j– αj j∀ 1 2 … k, , ,{ }.∈= fˆi0 j0 fˆi0 j0 a f i0 j0,( ) αjfi0 j0 j+, . j k–= k ∑= = a fˆi0 j0 fˆi0 j0 a f i0 j0,( ) αjfi0 j0 j+, , j λ–= m λ– 1– ∑= = λ 0 1 2 … m 1–, , , ,{ }.∈ a f i0 j0 a f i0 j0 a 1 21  7/21

The Formation of Operators with Given Properties to Solve Original Image Processing Tasks1

A. I. Novikov

Ryazan State Radio Engineering University, Ryazan, 390005 Russia email:

Abstract—The properties of operators with finite memory in terms of their application for measurement smoothing processing and detection of image brightnesschange boundaries are investigated. Examples of forming operators with given properties and examples of their application to processing real images are viewed.

Keywords: smoothing, an unbiased estimate, smoothing factor, brightness changes boundaries, boundaries detector, Gaussian blur.

DOI: 10.1134/S1054661815020194

Received in March 3, 2014




THE FORMATION OF OPERATORS WITH GIVEN PROPERTIES 231 2) operator A7,0 with mask = (–3,–2,– 1,0,1,2,3) will provide the smoothed estimation of partial derivative ; 3) operator A7,3 with the simplest set of weight fac tors = , 0, 0, 1, 0, 0, will provide the esti mation of second order partial derivative .

Herewith the first and the third operators are cen tral in the sense that estimation = ( , ) refers to the central point of the sliding window and for the second operator it refers to the first point.

Let’s consider properties of Am, λ operators and, having them as basis, let’s formulate the principles to build operators with given properties to solve the tasks of measurements smoothing and detection of bright nesschange boundaries.


One can regard the brightness function fij as an additive mixture of lowfrequency component zij (background) and random component ξij, i.e. (3)

The smoothing of original measurements is under stood as transformation providing the reduction of the mean square error of random component (ση < σξ). It is desirable that this reduction would be a maximum of (|ση – σξ| ). Also it is supposed that alongside smoothing operator Am, λ an unbiased estimation of the lowfre quency component should be provided. As a rule, this requirement contradicts the first.

Of all central operators A2k + 1, k the operator with mask (4) provides the best smoothing, having the length 2k + 1.

In this case ση = σξ if the values of random component ξ are not correlated.

Let’s estimate now the influence of operator

A2k + 1, k on lowfrequency component z. We will con sider that it is described adequately in the neighbor hood of the central point of the sliding window by a a 1 10  ∂f ∂x  ∂f ∂y ⎝ ⎠ ⎛ ⎞ a 1 2 –⎝ ⎛ 1 2 – ⎠ ⎞ ∂2f ∂x2  ∂ 2f ∂y2 ⎝ ⎠ ⎛ ⎞ fˆ a f fij zij ξi j.+= fˆi j Am λ, fij Am λ, zij Am λ, ξi j+ yij ηi j,+= = = a max a 1 2k 1+  1 2k 1+  … 1 2k 1+ , , ,⎝ ⎠ ⎛ ⎞ = 1 2k 1+  polynomial of some finite degree n. In practice, as a rule, n ≤ 3. It is easy to show that operator A2k + 1, k with mask (4) provides an unbiased estimation of polyno mials having a degree not more than the first. Opera tors Am, λ providing an unbiased estimation of polyno mials up to nth degree inclusive we will designate as with index n above.