180
Yu. Kotsiuba et al.
noise formation on the obtained phase map [3–5]. The mechanism of the speckle noise
formation and its nature are described in [1–3]. It should be noticed that phase decorrelation can have several sources in the DH [4]. It occurs when coherent light passes
through an inhomogeneous medium or being reflected from a rough surface. Both
cases take place during mechanical characteristics study of materials [1]. Speckle
noise can significantly distort the quality of the obtained phase image. As a result,
the spatial resolution, signal-to-noise ratio, and measurement accuracy decrease [1].
Therefore, it becomes impossible to carry out the unwrapping correctly.
Thus, it is necessary to carry out the noise filtering. The main task of filtering is
to increase the signal-to-noise ratio without removing the 2π phase gaps [6]. Today,
a lot of approaches to reduce the noise level are known [7–18]. All methods can be
divided into optical and digital. Optical methods are based on increasing the amount
of information received about the object by increasing the number of exposures. In
turn, the digital filtering methods using the mathematical algorithms extract information about the object which is obtained in a single exposure [7]. The main idea
of optical noise reduction techniques is the average of the several holograms which
were recorded under various object illumination conditions: wavelength [8], phase
[9], the angle of incidence [10], and the polarization [11]. Other optical techniques
for reducing speckle noise in a DH are based on the use of partially coherent light
sources [12].
In digital filtering methods, the image processing is usually done on the sines and
cosines images calculated from the phase to avoid the 2π phase jumps distortion on
the wrapped phase map [13, 14]. Today, several digital filtering methods for reducing
the speckle noise in DHI are known [7, 15–18], for example, wavelet filtering [16]
and filtering in the frequency domain using the Fourier transform [18].
In this work, we propose a new filter based on Chebyshev polynomials of the first
kind. The evaluation of the filtration errors is done using phase maps obtained as a
result of numerical simulation of the physical processes in the DHI.
2 Mathematical Description of a Filter Based
on Chebyshev Polynomials
This section describes the mathematical representation of Chebyshev polynomials
of the first kind, showing their application for filtering narrow peaks, as well as the
optimal number of polynomials for representing a function as a sum of polynomials.
Yu. Kotsiuba et al.
noise formation on the obtained phase map [3–5]. The mechanism of the speckle noise
formation and its nature are described in [1–3]. It should be noticed that phase decorrelation can have several sources in the DH [4]. It occurs when coherent light passes
through an inhomogeneous medium or being reflected from a rough surface. Both
cases take place during mechanical characteristics study of materials [1]. Speckle
noise can significantly distort the quality of the obtained phase image. As a result,
the spatial resolution, signal-to-noise ratio, and measurement accuracy decrease [1].
Therefore, it becomes impossible to carry out the unwrapping correctly.
Thus, it is necessary to carry out the noise filtering. The main task of filtering is
to increase the signal-to-noise ratio without removing the 2π phase gaps [6]. Today,
a lot of approaches to reduce the noise level are known [7–18]. All methods can be
divided into optical and digital. Optical methods are based on increasing the amount
of information received about the object by increasing the number of exposures. In
turn, the digital filtering methods using the mathematical algorithms extract information about the object which is obtained in a single exposure [7]. The main idea
of optical noise reduction techniques is the average of the several holograms which
were recorded under various object illumination conditions: wavelength [8], phase
[9], the angle of incidence [10], and the polarization [11]. Other optical techniques
for reducing speckle noise in a DH are based on the use of partially coherent light
sources [12].
In digital filtering methods, the image processing is usually done on the sines and
cosines images calculated from the phase to avoid the 2π phase jumps distortion on
the wrapped phase map [13, 14]. Today, several digital filtering methods for reducing
the speckle noise in DHI are known [7, 15–18], for example, wavelet filtering [16]
and filtering in the frequency domain using the Fourier transform [18].
In this work, we propose a new filter based on Chebyshev polynomials of the first
kind. The evaluation of the filtration errors is done using phase maps obtained as a
result of numerical simulation of the physical processes in the DHI.
2 Mathematical Description of a Filter Based
on Chebyshev Polynomials
This section describes the mathematical representation of Chebyshev polynomials
of the first kind, showing their application for filtering narrow peaks, as well as the
optimal number of polynomials for representing a function as a sum of polynomials.