CN103226813B - A kind of disposal route improving rainy day video image quality - Google Patents

A kind of disposal route improving rainy day video image quality Download PDF

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CN103226813B
CN103226813B CN201310106904.8A CN201310106904A CN103226813B CN 103226813 B CN103226813 B CN 103226813B CN 201310106904 A CN201310106904 A CN 201310106904A CN 103226813 B CN103226813 B CN 103226813B
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mrow
msubsup
function
raindrop
tone
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CN103226813A (en
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董蓉
包志华
罗磊
徐晨
周晖
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Nantong University Technology Transfer Center Co ltd
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Nantong University
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Abstract

The present invention relates to a kind of disposal route improving rainy day color video frequency image quality, comprise video image conversion, video raindrop detect and video raindrop are eliminated; Video image switch process comprise RGB image is converted to cone strength signal (L, M, S) after be converted to a colourless letter of transfer number and two opposite color letter of transfer numbers again; Detect at video raindrop and comprise according to pixels (x, y) place blocked by raindrop after color difference relationship detect raindrop and according to pixels (x, y) the colourless letter of transfer number in place a (x, y) raindrop are detected, video raindrop are eliminated and are comprised definition raindrop binary image B (x, y), to B (x, y) carry out Gaussian Blur and obtain G (x, y), by G (x, y) with B (x, y) weight matrix w (x, y) and by w (x, y), a is built new(x, y, r g new(x, y), r b new(x, y) inverse transformation obtains to RGB territory and removes image after rain.Advantage is, improves rain picture quality.

Description

Processing method for improving video image quality in rainy days
Technical Field
The invention relates to the technical field of machine vision and video image processing, in particular to a processing method for improving the quality of a video image in rainy days.
Background
However, the quality of a monitored image is often reduced due to the weather conditions such as rainfall and the like, so that monitoring personnel are prevented from finding hidden warning situations, the implementation of intelligent analysis algorithms such as target identification and the like is not facilitated, and the performance of the monitoring system is seriously influenced.
In the prior art, whether raindrops exist or not is judged by fitting linear ratio of brightness difference values of pixels in the same raindrop before and after being influenced by raindrops and background brightness, and then raindrops are removed in an averaging mode of pixels of two frames before and after the raindrops are adopted, but the raindrops are usually small in area and fitting is easily interfered by noise. Whether the raindrops exist in the image or not is generally judged according to the functional relation between the raindrop speed and the diameter and the threshold range of the rain drop length-width ratio in the perspective imaging model, meanwhile, the raindrop falling direction is obtained in a histogram mode, the rain drop elimination still adopts a mode of averaging pixels of two frames before and after, but the rain lines are often broken or merged due to noise, so that the real length-width ratio cannot be obtained. In addition, raindrops are detected from an RGB color domain, a background pixel brightness value is extracted by using a K-means clustering method to realize rain removal, but the whole video cannot be processed online in real time due to the requirement of processing the whole video. For raindrop characteristic analysis, raindrop characteristics are quantitatively analyzed in an HSV color space, and a measurement function for distinguishing a raindrop region and a moving object region is provided. For raindrop motion analysis, raindrops are taken as a motion foreground, raindrops are extracted by adopting a Gaussian background modeling method, and the method is only suitable for fixed scenes. In analyzing the characteristics of raindrop phenomenon in the frequency domain, a method of performing three-dimensional Fourier transform on a rainfall video sequence, detecting and eliminating raindrops in the frequency domain, and performing inverse transform on the raindrops to a video image is often adopted.
Therefore, the existing video raindrop detection method adopts a real number algorithm to process the brightness or color information of the image, and does not conform to the visual perception of human eyes; the existing video raindrop elimination algorithm is easy to cause edge mutation and image mosaic, so that an image after rain removal is not smooth, and if a method for performing smooth filtering on the image after rain removal is adopted, the image is fuzzy and information is lost.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a processing method for improving the video image quality in rainy days, which is in accordance with the visual perception of human eyes. The specific technical scheme is as follows:
the processing method comprises the following steps:
video image conversion: converting the RGB image into a cone intensity signal, and carrying out color logarithmic image processing on the cone intensity image to generate a logarithmic tone-free functionSignal and two logarithmically opposite tonesFunction(s)A signal, theThe signal passing through the inverse function of the isomorphic functionMapping to color tone function space to obtain corresponding colorless tone functionSignal and opponent tone functionA signal;
video raindrop detection: raindrops are detected through the change of the non-tone function signal of the same pixel position in at least three continuous frame images and the change of the difference value of the non-tone function before and after being shielded by raindrops, and the pixels shielded by the raindrops meet the following conditions:
the no-tone function signal varies as:
the contrast function difference is:
wherein:x、yis the plane coordinate value of the pixel point,nas the number of frames,a n-1 (x,y)、a n (x,y)、a n+1 (x,y) Are respectively asn-1、nn+1 no tone function corresponding to three consecutive frame images,r g n-1 (x,y)、r g n (x,y)、r g n+1 (x,y)、r b n-1 (x,y)、r b n (x,y)、r b n+1 (x,y) Corresponding opposite tone functions of three continuous frame images respectively,T a a threshold value is determined for a candidate raindrop,T Δ determining a threshold for false detection;
video raindrop elimination: binaryzation image of raindropsB(x,y) Implementing Gaussian blur acquisitionG(x,y) FromG(x,y) AndB(x,y) Constructing a weight matrixw(x,y) By means of a weight matrixw(x,y) And using preceding and succeeding frames of the successive framesNon-tone function of image recovery pixel shielded by raindropa new (x,y)And opponent tone functionr g new (x,y)、r b new (x,y)And obtaining the image without raindrops through inverse transformation to the RGB domain.
The processing method is further designed in that the conversion of the RGB image into the cone intensity image adopts the following linear transformation:
wherein:LMSrespectively representing the response of long-wave, medium-wave and short-wave visual cone perception cells in human eyes to incident light;RGBrepresenting the red, green and blue three-channel components of the incident light.
The processing method is further designed in that the color logarithmic image processing generates a logarithmic tone-free functionSignal and two logarithmic opponent tone functionsThe signal is obtained by the following procedure:
firstly, in the light absorption stage, the cone intensity signal is (L,M,S) Conversion to a color tone function (l,m,s) Signal:
then in the nonlinear response stage of the retinal nerve signals, using LIP isomorphic functionGenerating logarithmic tone function signals for each tone channel
Finally, in the opposite color generation stage, the logarithmic color tone function is converted into a logarithmic non-tone function by utilizing the linear chroma conversion matrixAnd two logarithmic opponent tone functions
In the above formula:C∈{L,M,S},c∈{l,m,s},l,m,sare respectivelyL,M,SThe corresponding color-tone function is calculated,C 0 for the purpose of reference to the luminance value,M 0 is an environment-dependent scale factor and is,in the form of an isomorphic function,are respectively provided withl,m,sCorresponding logarithm ofA color tone function.
The processing method is further designed in such a way that the raindrop binary image isB(x,y) Is defined as:
the processing method is further designed in that the pairsB(x,y)The gaussian blur of (a) is:
wherein:g(μ,σ)is a mean value ofμVariance ofσGaussian function of (1)
The processing method is further designed in thatG(x,y)AndB(x,y) Constructing a weight matrixw(x,y)Comprises the following steps:
the processing method is further designed in that the non-tone functiona new (x,y)And opponent tone functionr g new (x,y)r b new (x,y)Respectively as follows:
the invention innovatively applies a color logarithmic image processing theory to video raindrop detection, simulates three processes of light absorption of a cone receptive field, retina neural signal nonlinear response and opponent color signal generation in a human eye visual system, generates a non-tone function and two opponent tone functions, constructs a raindrop detection algorithm based on the three-channel function, and is more in line with human eye visual perception.
Drawings
FIG. 1 is a block flow diagram of the method of the present invention.
Fig. 2 is a comparison of the raindrop detection effect of CLIP operation according to the present invention and the detection effect of general brightness operation.
FIG. 3 is a comparison of the weighted rain removal and the average rain removal according to the present invention.
Fig. 4 shows the rain removing effect of the present invention.
Detailed Description
The method mainly comprises three steps of video image conversion, video raindrop detection and video raindrop elimination, and please refer to fig. 1.
In the video image conversion step:
firstly, RGB format images obtained by a common video acquisition device are converted into a cone intensity L, M, S signal in a human visual system, whereinLMSRespectively, the response of long-wave, medium-wave and short-wave cone perception cells in human eyes to incident light. This embodiment uses a linear transformation to implement:
(1)
in the above formulaRGBRespectively representing the red, green and blue three-channel components of the incident light.
For simulating the visual perception process of human eyes, the cone intensity signal (A)L,M,S) Performing Color Logarithmic Image Processing (CLIP) to convert color video image into a logarithmic non-tone functionSignal and two logarithmic opponent tone functionsA signal. The CLIP processing procedure of the present embodiment is specifically divided into three stages of light ray absorption, nonlinear response of retinal nerve signals, and opponent color generation. In the light absorption stage, the cone intensity signal (A)L,M,S) Conversion to a color tone function (l,m,s):
(2)
(2) In the formula:C∈{L,M,S},c∈{l,m,s},l,m,sare respectivelyL,M,SThe corresponding color-tone function is calculated,C 0 for the purpose of reference to the luminance value,M 0 is an environment-dependent scale factor.
In the nonlinear response stage of the retinal nerve signals, LIP isomorphic function is utilizedGenerating logarithmic tone function for each tone channel
(3)
(2) In the formula:are respectively provided withl,m,sCorresponding logarithmic color tone function.
In the opposite color generation stage, a logarithmic color tone function is converted into a logarithmic non-tone function by utilizing a linear chroma conversion matrixAnd two logarithmic opponent tone functions
(4)
Finally, willThe signal passing through the inverse function of the isomorphic functionMapping to color tone function space to obtain corresponding colorless tone functionSignal and opponent tone functionA signal. Obtained of (a,r g ,r b ) The three-channel color transfer function signal accords with the visual perception of human eyes and can highlight raindrop information when being used for raindrop detection.
In the video raindrop detection step:
because the raindrops can refract large-scale ambient light, the brightness of the scene after being shielded by the raindrops is often higher than the original background brightness of the scene, and on the other hand, the raindrops fall at a high speed, so that the same pixel position is rarely shielded by the raindrops at the same time in two adjacent frames, and the time-sequence brightness change of the same pixel position is in a jitter state. Therefore, at least three consecutive frame images are used, and the present embodiment uses three consecutive frame images. As a first condition of raindrop determination, a front-back change of a no-tone function signal of the same pixel position in consecutive frame images: for three continuous frames of imagesn-1、nn+1 frame without tone functiona n-1 (x,y)、a n (x,y)、a n+1 (x,y) The signal context should satisfy:
(5)
(6)
in the above two formulas:nas the number of frames,a n-1 (x,y)、a n (x,y)、a n+1 (x,y) Are respectively asn-1、nnNo tone function corresponding to +1 three continuous frame images; Θ is the subtraction of the LIP,T a for candidate raindrop judging threshold, the patent selects as: when in useM 0 =C 0 When the value of the electric field is not less than 255,T a and 10 is taken.
However, the expressions (5) and (6) only consider the luminance change characteristics of the raindrops, the detected raindrops may include noise, the motion foreground may also be detected, the raindrops detected by the expressions (5) and (6) may be referred to as raindrop candidates, and false detection may be further removed by combining with the opponent tone function as a second condition for raindrop determination. Considering that the difference between the brightness of the pixels blocked by raindrops in the continuous frames is:
(7)
wherein,I r is a pixel shielded by raindrops (x,y) The strength of (a) is high,I b is the background intensity of the pixel point and,xyis the plane coordinate value of the pixel point,for the irradiance caused by the raindrops,E b for the irradiance due to the background,the time when a pixel is occluded by a raindrop. Due to the fact that>>E b Therefore, the second term at the right end of equation (7) is negligible, and the raindrop refracts a large range of light, which often includes most of sky light, so that, from the color perspective, the color difference values of the front and back of the pixel shielded by the raindrop are approximately the same in each color channel, and in combination with the opposite tone function, the second condition that the pixel is determined as a raindrop is that:
(8)
thereinAre respectively as
(9)
(10)
Wherein,and | without CLIP Respectively for LIP multiplication, addition and modular operation,T Δ a threshold is determined for false detection.
Therefore, the pixels satisfying the expressions (5), (6), and (8) at the same time can be determined as raindrops.
In the video raindrop elimination step:
to remove raindrops from a video image, it is necessary to estimate pixel values of a background where a current frame is blocked by raindrops. Because the pixel is basically not shielded by raindrops in two adjacent frames, the average value of the pixel in the previous frame and the next frame can be used as the value of the current frame. However, due to the problem of camera focusing, raindrop edges in an image tend to be in a blurred state, and in addition, a gap sometimes exists in a raindrop detected due to noise influence, if only the pixel value of a detected raindrop part is restored, abrupt edges and image mosaics may be caused, and if a method of performing smooth filtering on the restored image is adopted, image blurring and information loss may be caused. In order to improve the image quality after rain removal, the invention adopts the following weighted reconstruction method to eliminate raindrops:
define the raindrop binary image asB(x,y):
(11)
For binary imageB(x,y)Obtaining images by Gaussian blur
(12)
Whereing(μ,σ)Is a mean value ofμVariance ofσIs constructed based onG(x,y)AndB(x,y) Weight matrixw(x,y)
(13)
By a weight matrixw(x,y) And restoring the raindrop-shielded pixels by using the front and rear frame images in the continuous frames, thereby obviously reducing the phenomena of edge mutation and mosaic, and eliminating the tone-free function signal of the raindrop-shielded pixel valuea new (x,y)
(14)
Will be described in the above formulaa n-1 (x,y)、a n (x,y)、a n+1 (x,y) Are respectively replaced byr g n-1 (x,y)、r g n (x,y)、r g n+1 (x,y) Can obtainr g new (x,y)In the above formulaa n-1 (x,y)、a n (x,y)、a n+1 (x,y) Are respectively replaced byr b n-1 (x,y)、r b n (x,y)、r b n+1 (x,y) Can obtainr b new (x,y)Then will (a new (x,y)、r g new (x,y)、r b new (x,y)) And inversely transforming to the RGB domain to obtain the image after rain removal.
The inventors of the present application made experiments as shown in fig. 2 to 4 using the method of the present invention and some existing raindrop detection and removal methods:
wherein, fig. 2 (a) is an original image before rain removal; graph (b) is the non-tonal difference of the candidate raindrop pixel and the background pixel after converting graph (a) to the CLIP non-tonal function; the graph (c) is the brightness difference value of the candidate raindrop pixel and the background pixel based on the brightness operation, and the comparison clearly shows that the result of the Color Logarithmic Image (CLIP) processing of the invention accords with the visual perception of human eyes, and compared with the general brightness operation, the raindrop information is more prominent.
FIG. 3 shows the rain removing effect of the present invention and the prior art, wherein (a) is the original drawing before rain removing, and (d) is the partial enlargement corresponding to the blue frame in (a); (b) the drawing is an effect drawing of removing rain by adopting the existing frame averaging method before and after addition, and the drawing (e) is a partial enlarged view corresponding to the blue frame in the drawing (b), and due to the defocusing effect of raindrops, traces still exist at the original raindrop edges after rain removal; (c) the effect graph of the weighted reconstruction method for removing rain is shown, the visible image quality is greatly improved, and the comparison shows the advantage of the weighted reconstruction method for removing rain.
Fig. 4 (a) is an original image before raining, and (b) is an effect image obtained by the raindrop removing method of the present invention, and shows a raindrop removing result of the algorithm of the present invention.

Claims (6)

1. A processing method for improving the quality of a color video image in rainy days is characterized by comprising the following steps:
video image conversion: converting the RGB image into a cone intensity signal, and carrying out color logarithmic image processing on the cone intensity image to generate a logarithmic tone-free functionSignal and two logarithmic opponent tone functionsA signal, theThe signal passing through the inverse function of the isomorphic functionMapping to the color tone function space to obtain corresponding non-color tone function a signal and anti-color tone function rg、rbA signal;
and a logarithmic tone-free function is generated by logarithmic image processingSignal and two logarithmic opponent tone functionsThe signal is obtained by the following procedure:
first, in the light absorption phase, the cone intensity signal (L, M, S) is converted into a color tone function (L, M, S) signal:
<math> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>:</mo> <mi>C</mi> <mo>&RightArrow;</mo> <mi>c</mi> <mo>=</mo> <msub> <mi>M</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mi>C</mi> <msub> <mi>C</mi> <mn>0</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>
then in the nonlinear response stage of the retinal nerve signals, using LIP isomorphic functionGenerating logarithmic tone function signals for each tone channelFinally, in the opposite color generation stage, the logarithmic color tone function is converted into a logarithmic non-tone function by utilizing the linear chroma conversion matrixAnd two logarithmic opponent tone functions
a ~ r ~ g r ~ b = 40 / 61 20 / 61 1 / 61 1 - 12 / 11 1 / 11 1 / 9 1 / 9 - 2 / 9 l ~ m ~ s ~ ;
Video raindrop detection: raindrops are detected through the change of the non-tone function signal of the same pixel position in at least three continuous frame images and the change of the difference value of the non-tone function before and after being shielded by raindrops, and the pixels shielded by the raindrops meet the following conditions:
the no-tone function signal varies as:
an(x,y)<an-1(x,y)&an(x,y)<an+1(x,y),
|an(x,y)Θan-1(x,y)|>Ta&|an(x,y)Θan+1(x,y)|>Ta
the contrast function difference is:
<math> <mrow> <msub> <mi>&Delta;r</mi> <mi>g</mi> </msub> <mo>=</mo> <mo>(</mo> <msubsup> <mi>r</mi> <mi>g</mi> <mi>n</mi> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>&Theta;</mi> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&CircleTimes;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>g</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CirclePlus;</mo> <msubsup> <mi>r</mi> <mi>g</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>)</mo> <mo>)</mo> <mo>,</mo> </mrow> </math>
<math> <mrow> <msub> <mi>&Delta;r</mi> <mi>b</mi> </msub> <mo>=</mo> <mo>(</mo> <msubsup> <mi>r</mi> <mi>b</mi> <mi>n</mi> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>&Theta;</mi> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&CircleTimes;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>b</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CirclePlus;</mo> <msubsup> <mi>r</mi> <mi>b</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>)</mo> <mo>)</mo> <mo>,</mo> </mrow> </math>
|ΔrgΘΔrb|CLIP<TΔ
video raindrop elimination: g (x, y) is obtained by implementing Gaussian blur on the raindrop binary image B (x, y), a weight matrix w (x, y) is constructed by the G (x, y) and the B (x, y), and the non-color-tone function a of the raindrop-sheltered pixels is recovered by the weight matrix w (x, y) and the front and rear frame images in the continuous framesnew(x, y) and the inverse tone function rg new(x,y)、rb new(x, y) obtaining an image without raindrops through inverse transformation to an RGB domain;
wherein: c belongs to { L, M, S }, C belongs to { L, M, S }, L, M, S are color tone functions corresponding to L, M, S respectively, and C is the color tone function corresponding to L, M, S respectively0For reference luminance value, M0Is an environment-dependent scale factor and is,in the form of an isomorphic function,logarithmic color tone function corresponding to l, m and s, x and y being plane coordinate values of pixel points, n being frame number, an-1(x,y)、an(x,y)、an+1(x, y) are respectively the no tone function corresponding to three continuous frame images of n-1, n and n +1, rg n-1(x,y)、rg n(x,y)、rg n+1(x,y)、rb n-1(x,y)、rb n(x,y)、rb n+1(x, y) are the corresponding opposite tone functions of three continuous frame images, TaDetermining a threshold, T, for a candidate raindropΔA threshold is determined for false detection.
2. The method as claimed in claim 1, wherein the converting of the RGB image into the cone intensity image is performed by a linear transformation as follows:
L M S = 0.3634 0.6102 0.0264 0.1246 0.8138 0.0616 0.000 0.0602 0.9389 R G B ,
wherein: l, M, S represent responses of long, medium, and short wavelength cones sensing cells in the human eye to incident light, respectively; r, G, B represent the red, green, and blue three-channel components of the incident light.
3. The method as claimed in claim 2, wherein the raindrop binarization processing method is applied to the color video image in rainy daysThe image B (x, y) is defined as:
4. a process of improving the quality of color video images in rainy weather as claimed in claim 2, wherein said gaussian blur for B (x, y) is: g (x, y) ═ B (x, y) × G (μ, σ);
wherein: g (μ, σ) is a gaussian function with mean μ and variance σ.
5. A process of improving the quality of color video images in rainy weather as claimed in claim 2, wherein G (x, y) and B (x, y) construct a weight matrix w (x, y) as: w ( x , y ) = { G ( x , y ) , i f B ( x , y ) = 0 B ( x , y ) , i f B ( x , y ) = 255 .
6. the method of claim 2, wherein said non-tonal function a is used to enhance the quality of color video images in rainy weathernew(x, y) and the opponent tone function rg new(x,y)、rb new(x, y) are respectively:
<math> <mrow> <msup> <mi>a</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>w</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&CircleTimes;</mo> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CirclePlus;</mo> <msup> <mi>a</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>)</mo> <mo>&CirclePlus;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>w</mi> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <msup> <mi>a</mi> <mi>n</mi> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>
<math> <mrow> <msubsup> <mi>r</mi> <mi>g</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>w</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&CircleTimes;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>g</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CirclePlus;</mo> <msubsup> <mi>r</mi> <mi>g</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>)</mo> <mo>&CirclePlus;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>w</mi> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <msubsup> <mi>r</mi> <mi>g</mi> <mi>n</mi> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>
<math> <mrow> <msubsup> <mi>r</mi> <mi>b</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>w</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&CircleTimes;</mo> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>b</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&CirclePlus;</mo> <msubsup> <mi>r</mi> <mi>b</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>)</mo> <mo>&CirclePlus;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>w</mi> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <msubsup> <mi>r</mi> <mi>b</mi> <mi>n</mi> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </math>
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