CN109242799B - Variable-threshold wavelet denoising method - Google Patents

Variable-threshold wavelet denoising method Download PDF

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CN109242799B
CN109242799B CN201811094442.1A CN201811094442A CN109242799B CN 109242799 B CN109242799 B CN 109242799B CN 201811094442 A CN201811094442 A CN 201811094442A CN 109242799 B CN109242799 B CN 109242799B
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CN109242799A (en
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赵佰亭
王风
郭永存
贾晓芬
黄友锐
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Anhui University of Science and Technology
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Abstract

The invention discloses a wavelet denoising method with a variable threshold, which comprises five steps. Inputting an original image and adding corresponding Gaussian noise; step two, selecting a wavelet basis function and determining the layer number O of wavelet decomposition: decomposing the noise image S to obtain a first-layer low-frequency coefficient A1, horizontal and vertical high-frequency coefficients H1 and V1 and a diagonal high-frequency coefficient D1; decomposing A1 to obtain a second layer low-frequency coefficient A2, horizontal and vertical high-frequency coefficients H2 and V2 and a diagonal high-frequency coefficient D2; decomposing A2 to obtain a third layer of low-frequency coefficient A3, horizontal and vertical high-frequency coefficients H3 and V3 and a diagonal high-frequency coefficient D3; sequentially decomposing the mixture to an O-th layer; step three, selecting a combined wavelet threshold and a straight line
Figure DDA0001805199580000011
Processing the wavelet coefficients for a wavelet threshold function of the asymptote line; step four, performing wavelet reconstruction on the wavelet coefficient after threshold quantization; and fifthly, outputting the denoised image. The invention can improve the precision of wavelet transform processing noise signals, effectively improve the denoising effect of images and obtain high-quality denoised images.

Description

Variable-threshold wavelet denoising method
Technical Field
The invention relates to the field of denoising in digital image processing, in particular to a wavelet denoising method with a variable threshold value.
Background
The image is susceptible to noise in the processes of forming, recording, processing and transmitting, so that the image quality is reduced, the image is blurred, and even the image characteristics are submerged, which brings difficulties to subsequent image region segmentation, target identification, edge extraction and the like. Therefore, removing noise is a key preprocessing step before processing the image. In order to obtain high quality digital images, it is necessary to perform noise reduction on the images to remove unwanted information from the signal while maintaining the integrity (i.e., the main features) of the original information as much as possible. Therefore, the noise reduction processing has been a hot spot of the image processing.
The wavelet denoising is generally divided into three types, wherein the first type is image denoising by using a wavelet transform modulus maximum method; the second type is that after wavelet transformation is carried out on a signal containing noise, correlation of wavelet coefficients between adjacent scales is calculated, the wavelet coefficients are subjected to accepting and rejecting according to the magnitude of the correlation, and finally, the de-noised signal is obtained through reconstruction; the third type is threshold denoising, which is realized by performing different processing on wavelet coefficients according to different wavelet coefficient distributions of signals and noise after wavelet transformation. The wavelet threshold denoising method is small in operation amount, simple to implement and widely applied.
In the patent 'DR image denoising method and system based on wavelet transform' (patent number: 102663695A), an improved soft threshold function is adopted to process an image, wavelet reconstruction is carried out to obtain a high-frequency coefficient and a low-frequency coefficient of a first layer, the high-frequency coefficient of the first layer is processed by adopting a hard threshold function, and then reconstruction is carried out again. The method organically combines the soft threshold function and the hard threshold function together, obtains better denoising effect, improves the signal-to-noise ratio of the image, but still cannot solve the defects of the soft threshold function and the hard threshold function.
Compared with a DR image denoising method and system based on wavelet transformation, the method has the advantages that:
(1) wavelet threshold function of the invention is represented by a straight line
Figure BDA0001805199560000011
The method solves the problem of edge mode caused by the discontinuity of the hard threshold function at the threshold point and the constant deviation between the soft threshold functions for the gradual lineThe problem of paste.
(2) The wavelet threshold provided by the invention has self-adaptability and meets the characteristics that the noise signal of the image signal is continuously reduced and the image signal is continuously increased in the decomposition process. When the number of decomposition layers i is 1, the threshold value is continuously decreased as the number of decomposition layers increases, as in the case of the global threshold value.
(3) The wavelet threshold value provided by the invention can be adaptively updated according to different image decomposition layer numbers, so that the wavelet threshold value is adaptive to the characteristics of wavelet transform coefficients of each layer.
The invention aims to provide a wavelet threshold and a wavelet threshold function, which can improve the precision of processing a noise signal by wavelet transformation, effectively improve the denoising effect of an image and obtain a high-quality denoised image.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a variable-threshold wavelet denoising method to improve the denoising effect of an image.
The invention relates to a wavelet denoising method with a variable threshold, which is characterized by comprising the following steps:
inputting an original image and adding corresponding Gaussian noise;
selecting a proper wavelet basis function, determining the layer number O of wavelet decomposition, and then performing O-layer wavelet decomposition on the noisy image:
(a) decomposing the noise image S to obtain a first-layer low-frequency coefficient A1, a first-layer horizontal high-frequency coefficient H1, a first-layer vertical high-frequency coefficient V1 and a first-layer diagonal high-frequency coefficient D1;
(b) decomposing the first layer low-frequency coefficient A1 to obtain a second layer low-frequency coefficient A2, a second layer horizontal high-frequency coefficient H2, a second layer vertical high-frequency coefficient V2 and a second layer diagonal high-frequency coefficient D2;
(c) decomposing the second layer low-frequency coefficient A2 to obtain a third layer low-frequency coefficient A3, a third layer horizontal high-frequency coefficient H3, a third layer vertical high-frequency coefficient V3 and a third layer diagonal high-frequency coefficient D3;
(d) sequentially decomposing the low-frequency coefficients of the current layer until the required number of wavelet decomposition layers of the O layers is reached;
step three, threshold quantization of the wavelet coefficient, selecting proper wavelet threshold and wavelet threshold function to process the wavelet coefficient;
step four, wavelet coefficient reconstruction, namely performing wavelet reconstruction on the wavelet coefficient after threshold quantization;
and fifthly, outputting the denoised image.
Further, the wavelet denoising method with variable threshold according to claim 1, wherein in step three, the threshold of the wavelet coefficient is quantized, and the wavelet coefficient is processed by selecting a proper wavelet threshold and a wavelet threshold function, where the wavelet threshold function is,
Figure BDA0001805199560000031
in the formula (1), m is a positive number,
Figure BDA0001805199560000033
for the processed wavelet coefficients, Wl，kIs a wavelet coefficient, T is a wavelet threshold
Furthermore, the method for denoising by using wavelet with variable threshold is characterized in that in the third step, the threshold of the wavelet coefficient is quantized, and the wavelet coefficient is processed by selecting a proper wavelet threshold and a wavelet threshold function, wherein the wavelet threshold function is a straight line
Figure BDA0001805199560000034
Is an asymptotic line.
Furthermore, the method for denoising by wavelet with variable threshold is characterized in that, in the third step, the threshold of the wavelet coefficient is quantized, and the wavelet coefficient is processed by selecting a proper wavelet threshold and a wavelet threshold function, wherein the wavelet threshold T is,
Figure BDA0001805199560000035
in the formula (2), i is the number of layers of the current decomposition, σ is the signal variance, M × N is the size of the high-frequency coefficient of the ith layer, and the threshold value can be reduced along with the increase of the number of the layers of the decomposition, so that the method conforms to the change of a noise signal and an image signal in the practical process of wavelet decomposition.
Furthermore, the method for denoising by using a variable threshold is characterized in that in the third step, the best image denoising effect can be obtained when m is greater than or equal to 50 and less than or equal to 60 in the wavelet threshold function.
Compared with the prior art, the invention has the following technical effects:
(1) wavelet threshold function of the invention is represented by a straight line
Figure BDA0001805199560000036
Is an asymptote line and is continuous at the position +/-T of the threshold value point, not only the defect that the hard threshold value function is discontinuous at the threshold value point is solved, but also the soft threshold value function can be solved
Figure BDA0001805199560000037
And Wj,kThere is a problem of edge blurring due to a constant deviation therebetween. The wavelet threshold function of equation (1) is demonstrated below to
Figure BDA0001805199560000038
Is an asymptotic line.
When W isl，k→T+And Wl，k→T-When the temperature of the water is higher than the set temperature,
Figure BDA0001805199560000039
when W isl，k→-T+And Wl，k→-T+Time of flight
Figure BDA00018051995600000310
Therefore, the improved threshold function is continuous at the threshold point +/-T, and the visual distortion phenomena of pseudo Gibbs effect, ringing and the like of the denoised image caused by the discontinuity of the hard threshold function at the threshold point +/-T are solved.
Is provided with
Figure BDA00018051995600000311
Wherein m and T are constants
Because:
Figure BDA0001805199560000032
Figure BDA0001805199560000041
Figure BDA0001805199560000042
Figure BDA0001805199560000043
therefore, the function f (x) is an asymptote line with y ═ x, and the wavelet threshold function of the formula (1) is a straight line
Figure BDA0001805199560000044
Is an asymptotic line. The problem that the wavelet coefficient processed by the soft threshold function has constant deviation with the original wavelet coefficient is solved, and the processing effect of the image on the edge details is closer.
(2) The wavelet threshold function of the invention is continuous at the threshold point +/-T, thus solving the visual distortion phenomena of pseudo Gibbs effect, ringing and the like of the denoised image caused by the discontinuity of the hard threshold function at the threshold point +/-T.
(3) The wavelet threshold provided by the invention has self-adaptability and meets the characteristics that the noise signal of the image signal is continuously reduced and the image signal is continuously increased in the decomposition process.
(4) The wavelet threshold provided by the invention can solve the problem of the conventional global threshold, namely ln (e + 2) in formula (2)(1-i)-1) corresponds to a puncturing factor, and when the number of decomposition layers i is 1, the threshold value is continuously decreased as the number of decomposition layers increases, as with the global threshold value.
(5) The wavelet threshold value provided by the invention can be adaptively updated according to different image decomposition layer numbers, so that the wavelet threshold value is adaptive to the characteristics of wavelet transform coefficients of each layer.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a wavelet hard threshold function image;
FIG. 2 is a wavelet soft threshold function image;
FIG. 3 is a threshold function image of the present invention;
FIG. 4 is a flow chart of a variable threshold wavelet denoising method;
FIG. 5 is a schematic diagram of a wavelet decomposition process;
FIG. 6 is a flow chart of wavelet coefficient threshold quantization;
FIG. 7 is a denoising result of a Lena image with a noise of 10 gauss added by different methods;
FIG. 8 is a denoising result of a Lena image with different methods for adding 20 Gaussian noise to the Lena image;
FIG. 9 is a denoising result of a different method for adding 30 Gaussian noise to a Lena image;
wherein S is a noise image; a1, A2 and A3 are respectively a first layer, a second layer and a third layer of low-frequency coefficients; g1, G2 and G3 are respectively a first layer, a second layer and a third layer of high-frequency coefficients; g11, G22 and G33 are respectively the first, second and third layers of high-frequency coefficients after threshold quantization; the high frequency coefficients G for each layer include horizontal, vertical and diagonal high frequency coefficients H, V and D.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Fig. 1 and 2 are hard and soft threshold function graphs of wavelets, respectively, when the wavelets are used for image denoising, the reconstructed image has visual distortions such as pseudo-gibbs effect and ringing due to the fact that the hard threshold function is discontinuous at a threshold point. However, the wavelet signal processed by the soft threshold function has a constant deviation with the wavelet signal of the original image, which causes a reduction in accuracy during reconstruction and blurs the edge of the image. To improve the above problem, the present invention designs the threshold function of fig. 3.
As shown in fig. 4, the present invention discloses a wavelet denoising method with variable threshold, which comprises five steps.
Step S1, inputting an original image and adding corresponding Gaussian noise;
step S2, selecting wavelet basis functions and determining the number of layers O of the wavelet decomposition: decomposing the noise image S according to the wavelet decomposition diagram of FIG. 5 to obtain a first-layer low-frequency coefficient A1, horizontal and vertical high-frequency coefficients H1 and V1, and a diagonal high-frequency coefficient D1; decomposing A1 to obtain a second layer low-frequency coefficient A2, horizontal and vertical high-frequency coefficients H2 and V2 and a diagonal high-frequency coefficient D2; decomposing A2 to obtain a third layer of low-frequency coefficient A3, horizontal and vertical high-frequency coefficients H3 and V3 and a diagonal high-frequency coefficient D3; sequentially decomposing the mixture to an O-th layer;
step S3, selecting a combined wavelet threshold and a straight line
Figure BDA0001805199560000051
Processing the wavelet coefficients for a wavelet threshold function of the asymptote line;
step S4, quantizing the threshold according to the wavelet coefficient threshold quantization flowchart of fig. 6, and then performing wavelet reconstruction on the quantized wavelet coefficients;
and step S5, outputting the denoised image.
Further, the wavelet denoising method with variable threshold according to claim 1, wherein in step three, the threshold of the wavelet coefficient is quantized, and the wavelet coefficient is processed by selecting a proper wavelet threshold and a wavelet threshold function, where the wavelet threshold function is,
Figure BDA0001805199560000061
wherein m is a positive number, and m is a positive number,
Figure BDA0001805199560000063
for the processed wavelet coefficients, Wl，kIs the wavelet coefficient and T is the wavelet threshold.
Furthermore, the method for denoising by using wavelet with variable threshold is characterized in that in the third step, the threshold of the wavelet coefficient is quantized, and the wavelet coefficient is processed by selecting a proper wavelet threshold and a wavelet threshold function, wherein the wavelet threshold function is a straight line
Figure BDA0001805199560000064
Is an asymptotic line.
Furthermore, the method for denoising by wavelet with variable threshold is characterized in that, in the third step, the threshold of the wavelet coefficient is quantized, and the wavelet coefficient is processed by selecting a proper wavelet threshold and a wavelet threshold function, wherein the wavelet threshold T is,
Figure BDA0001805199560000062
wherein i is the number of layers of the current decomposition, σ is the signal variance, M × N is the size of the high-frequency coefficient of the ith layer, and the threshold can be reduced along with the increase of the number of the layers of the decomposition, so that the method conforms to the change of noise signals and image signals in the practical process of wavelet decomposition.
Furthermore, the method for denoising by using a variable threshold is characterized in that in the third step, the best image denoising effect can be obtained when m is greater than or equal to 50 and less than or equal to 60 in the wavelet threshold function.
To verify the effectiveness of the present invention, simulation experiments were performed.
The experiments were all programmed in the environment of MATLAB 2018a, running on a PC configured as intel (R) core (TM) i5-5200U CPU2.19GHz. In the experimental process, the number of layers of wavelet decomposition is N-3, and m in the formula (1) is 58.
After gaussian noise with the resolution of 512 × 512 is added to a standard Lena image, the gaussian noise has the σ of 10, the σ of 20, and the σ of 30, a hard threshold function, a soft threshold function, a combination of a soft threshold function and a soft threshold function (patent 'DR image denoising method and system based on wavelet transform', patent number: 102663695a) and the method provided by the present invention are respectively adopted to denoise, and the effect after denoising is as shown in fig. 7, fig. 8, and fig. 9. The peak signal-to-noise ratio (PSNR) is used to measure the denoising effect, and the denoising result is shown in table 1.
TABLE 1 De-noising result comparison
Figure BDA0001805199560000071
As can be seen from the test results in Table 1, the denoising method of the present invention can obtain better PSNR.
The above description is only an example of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (3)

1. A wavelet denoising method with a variable threshold is characterized by comprising the following steps:
inputting an original image and adding corresponding Gaussian noise;
step two, selecting a wavelet basis function, determining the layer number O of wavelet decomposition, and then performing O-layer wavelet decomposition on the noisy image:
(a) decomposing the noise image S to obtain a first-layer low-frequency coefficient A1, a first-layer horizontal high-frequency coefficient H1, a first-layer vertical high-frequency coefficient V1 and a first-layer diagonal high-frequency coefficient D1;
(b) decomposing the first layer low-frequency coefficient A1 to obtain a second layer low-frequency coefficient A2, a second layer horizontal high-frequency coefficient H2, a second layer vertical high-frequency coefficient V2 and a second layer diagonal high-frequency coefficient D2;
(c) decomposing the second layer low-frequency coefficient A2 to obtain a third layer low-frequency coefficient A3, a third layer horizontal high-frequency coefficient H3, a third layer vertical high-frequency coefficient V3 and a third layer diagonal coefficient D3;
(d) sequentially decomposing the low-frequency coefficients of the current layer until the required number of wavelet decomposition layers of the O layers is reached;
step three, threshold quantization of the wavelet coefficient, selecting a wavelet threshold and a wavelet threshold function to process the wavelet coefficient;
in the third step, the threshold value of the wavelet coefficient is quantized, and the wavelet threshold value and the wavelet threshold function are selected to process the wavelet coefficient, wherein the wavelet threshold function is,
Figure FDA0003220532720000011
in the formula (1), m is a positive number,
Figure FDA0003220532720000012
for the processed wavelet coefficients, Wl,kIs a wavelet coefficient, T is a wavelet threshold;
in the third step, the threshold value of the wavelet coefficient is quantized, the wavelet threshold value and the wavelet threshold value function are selected to process the wavelet coefficient, the wavelet threshold value T is,
Figure FDA0003220532720000013
in the formula (2), i is the number of layers of current decomposition, sigma is the signal variance, M × N is the size of the high-frequency coefficient of the ith layer, and the threshold can be reduced along with the increase of the number of the layers of decomposition, so that the method conforms to the change of noise signals and image signals in the practical process of wavelet decomposition;
step four, wavelet coefficient reconstruction, namely performing wavelet reconstruction on the wavelet coefficient after threshold quantization;
and fifthly, outputting the denoised image.
2. The wavelet denoising method of claim 1, wherein in step three, the threshold quantization of the wavelet coefficients, selecting the wavelet threshold and the wavelet threshold function to process the wavelet coefficients, the wavelet threshold function being a straight line
Figure FDA0003220532720000021
Is an asymptotic line.
3. The method as claimed in claim 1, wherein in step three, the best image denoising effect can be obtained when m is greater than or equal to 50 and less than or equal to 60 in the wavelet threshold function.
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