RU60242U1 - DEVICE FOR CALCULATING A CONTINUOUS TWO-DIMENSIONAL WAVELET TRANSFORM WITH AN ARBITRARY FILTER ANGLE - Google Patents

Abstract

The device relates to automation and computer technology and can be used in digital processing systems of two-dimensional signals to obtain a wavelet image of the signal under study. In particular, the proposed device can be used to analyze images, identify features, periodic dependencies and local heterogeneities. The utility model is aimed at obtaining a frequency-spatial representation of a two-dimensional signal using a continuous two-dimensional wavelet transform. The specified technical result is achieved by calculating a two-dimensional wavelet filter with an arbitrary choice of scaling factors and rotation angle. For this, two blocks for calculating the values of the functions sin and cos, five multipliers, two adders and a block for calculating the values of the wavelet filter, which provide the calculation of the wavelet filter rotated by an arbitrary angle Θ, are added to the proposed device. The obtained wavelet filter is stored in random access memory and used to calculate the convolution of the analyzed signal in the device for calculating two-dimensional convolution.

Description

The device relates to automation and computer technology and can be used in digital processing systems of two-dimensional signals to obtain a wavelet image of the signal under study.

In particular, the proposed device can be used to analyze images, identify features, periodic dependencies and local heterogeneities.

Since in the computer memory two-dimensional signals are represented as matrices of instantaneous values, i.e. in discrete form, then to obtain a wavelet image of such signals, an analogue of the continuous wavelet transform for discrete signals should be used. For one-dimensional signals, it is written as

,

where s is the signal, ψ is the wavelet, a is the scaling factor, and b is the wavelet shift.

The wavelet transform of discrete signals is equivalent to the convolution of a signal with some filter. Usually, a pair of filters is used to decompose the signal along the basis of wavelet functions: low-frequency (or scaling function) φ and high-frequency (or wavelet) ψ [3]. The presence of a pair of filters for implementing the wavelet transform is inherent, as a rule, only to orthogonal wavelets. Whereas, bases based on continuous wavelets, as a rule, are not strictly orthonormal, and bases that have only stability and “approximate” orthogonality properties are often used [1]. Therefore, to calculate the continuous wavelet transform, only the high-pass filter ψ is used.

To generalize the one-dimensional wavelet transform to the two-dimensional case, various approaches are used. One of these approaches is the choice of the wavelet ψ∈L ² (R) with or without spherical symmetry and rotation input, along with shifts and contractions. In this case, the wavelet will be defined as [2]

where Θ is the angle of rotation of the filter, R _Θ is the rotation matrix, which has the form

.

In the calculations there is no need to calculate the wavelets for all possible shifts b ₁ and b ₂ . It is enough to calculate the wavelet at b ₁ = b ₂ = 0 and use it for convolution. In this case (1) takes the form

A common example of a wavelet used to calculate a continuous wavelet transform is the “Mexican hat” [4], for the two-dimensional case this type of wavelet is written as:

where h (x, y), taking into account (2), has the form

.

The rotation of the wavelet by the angle Θ is carried out as

ψ (x, y, Θ) = ψ (xcosΘ-ysinΘ, xsinΘ + ycosΘ).

Using this approach to obtain a two-dimensional wavelet filter allows you to design filters with different compression ratios in two directions and the rotation angle. This allows one to better detect local inhomogeneities of two-dimensional signals that have an irregular shape and are rotated by an arbitrary angle.

In this case, a two-dimensional continuous wavelet transform of a signal represented by a matrix of instantaneous values can be obtained as a result of a two-dimensional convolution of the signal with a filter.

A known method of series-parallel wavelet transform [Copyright certificate RU No. 22489850, IPC 7 G 06 F 17/14, BI No. 10, 2005], which consists in using a pair of filters for wavelet decomposition of the original signal specified in discrete time samples (Mall scheme) .

A device for the fast calculation of discrete wavelet transform

a signal with an arbitrary step of discretization of scale factors [Copyright certificate RU No. 2246132, IPC 7 G 06 F 17/14, BI No. 4, 2005], based on the method of calculating the continuous wavelet transform by means of the scalar product of the signal under study and the basic functions.

A common disadvantage of these devices is the inability to calculate a two-dimensional wavelet transform.

A device for calculating two-dimensional convolution [Author's certificate RU No. 2042209, IPC 6 G 06 T 1/00, BI No. 23, 1995], which allows to calculate the convolution of a two-dimensional signal input to the device with a filter supplied to the second input of the device.

This device is selected as a prototype.

The technical result of the proposed device is the ability to calculate a continuous two-dimensional wavelet transform for discrete signals.

The specified technical result is achieved by calculating the wavelet filter with an arbitrary choice of scaling factors and rotation angle.

The figure shows a diagram of the proposed device. The device consists of a control device - block 1, devices for calculating the values of the functions sin and cos (SIN, COS) - blocks 2 and 3, respectively, five multipliers - blocks 4-8, two adders - blocks 9 and 10, a block for calculating the values of the wavelet filter ( BVF) according to the formula (3) - block 11, random access memory (RAM) - block 12, analog-to-digital converter (ADC) - block 13 and two-dimensional convolution calculation device - block 14.

The principle of operation of the proposed device is as follows. From the control device, the angle of rotation угла of the wavelet filter is supplied to the inputs of the SIN and COS blocks, the outputs of which are connected to the second inputs of the multipliers 4, 5 and 6, 7, respectively, the first inputs of which from the control device are the indices x and y of the wavelet filter elements. The output of the multiplier 5 is connected to the input of the multiplier 8, in which the input value is multiplied by (-1). The outputs of the multipliers 8 and 6 are connected to the input of the adder 9, at the output of which an index x 'of the wavelet filter element is rotated through an angle Θ, and the output of the adder 10, the inputs of which are connected to the outputs of the multipliers 4 and 7, is the index y' of the element

filter.

The outputs of the adders are connected to the first two inputs of the BCF, the second two inputs of which from the control device receive the values of the scaling factors a ₁ and a ₂ . At the output of the BVF, the values of the wavelet filter are rotated by an angle Θ, which are stored in RAM.

From RAM, the matrix of wavelet values is fed to the first input of a two-dimensional convolution device, the second input of which from the ADC output receives a discrete sample of samples s (x, y) of the analyzed signal s (t ₁ , t ₂ ) received at the ADC input. At the output of the convolution device, a discrete sample of the converted signal W (a ₁ , a ₂ ) is formed, which is a set of wavelet transform coefficients.

Information sources

1. Vorobiev V.I., Gribunin V.G. Theory and practice of wavelet transform. - S.-Pb .: VUS, 1999 .-- 204 p.

2. Finish I. Ten lectures on wavelets. - Izhevsk: Research Center "Regular and chaotic dynamics", 2001. - 464 p.

3. Malla S. Wavelets in signal processing: Trans. from English - M .: Mir, 2005 .-- 671 p., Ill.

4. Pereberin A. V. On the systematization of wavelet transforms // Computational methods and programming. - 2001 - T.2. - S.15-40

Claims (1)

A device for calculating a continuous two-dimensional wavelet transform with an arbitrary angle of rotation of the filter, performing two-dimensional convolution of the analyzed signal with a wavelet filter and containing blocks for calculating two-dimensional convolution, characterized in that it contains blocks for calculating the values of the functions sin and cos, the input of which is output the control device receives the value of the angle of rotation Θ of the filter, and the outputs of the blocks are connected to the second inputs of the 1st, 2nd and 3rd, 4th multipliers, respectively, to the first inputs of which from the device the indices x and y of the elements of the wavelet filter, and the index x goes to the inputs of the 1st and 3rd multipliers, and the index y goes to the inputs of the 2nd and 4th multipliers, and the output of the 2nd multiplier is connected to the first the input of the 5th multiplier, the second input of which receives the value (-1), and the output, together with the output of the 3rd multiplier, is connected to the inputs of the 1st adder, from the output of which the first input of the block for calculating the values of the wavelet filter (BVF) index x 'of the filter element, and the second input receives the index y' of the filter element from output 2- of the adder, the inputs of which are connected to the outputs of the 1st and 4th multipliers, in addition, the values of the scaling factors a ₁ and a ₂ are received at the 3rd and 4th inputs of the BCF, and the values of the wavelet filter elements are generated at the output, rotated by an angle Θ, which is stored in random access memory, from the output of which a wavelet filter is fed to the first input of the device for calculating two-dimensional convolution, and a discrete sample of samples of the analyzed signal comes to the second input from the output of the analog-to-digital converter (ADC), upivshego the ADC input and the output of the convolution device formed wavelet transform coefficients.