# Simulation of Video Image Fault Tolerant Coding Transmission in Digital Multimedia

Authors: Feng Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to effectively improve the quality of video image transmission, this paper proposes a method of digital multimedia video image coding. The transmission of digital multimedia video image fault-tolerant coding requires sparse decomposition of a digital multimedia video image to obtain the linear form of the image and complete the transmission of video image fault-tolerant coding. The traditional method of fault-tolerant coding is based on human visual characteristics but ignores the linear form of the digital multimedia video image, which leads to the unsatisfactory effect of coding and transmission. In this paper, a fault-tolerant coding method based on wavelet transform and vector quantization is proposed to decompose and reconstruct digital multimedia video images. The smoothness of wavelet transform can remove visual redundancy; the decomposed image is vector quantized. The mean square deviation method and the similar scalar optimal quantization method are used to select and calculate the image vector, construct the over complete database of a digital multimedia video image, and normalize it; the digital multimedia video image is thinly decomposed by asymmetric atoms, and a linear representation of the image is obtained. According to the above-given operations, we can master the distribution range and law of pixels and realize fault-tolerant coding. The experimental results show that when the number of iterations is 15, the CR index is the same, PSNR increases by 8.7%, coding is 23.7% faster and decoding is 15% faster.Conclusion. The proposed method can not only improve the speed of fault-tolerant coding but also improve the quality of video image transmission.


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## Body

## 1. Introduction

At present, video images have greatly affected people’s life. People obtain visual perception through video and also use images to serve themselves. With the development of image recognition technology, this technology has been widely used in video images. Through image recognition, people can get more concerned objects from video. An intelligent video surveillance system is a carrier of image recognition technology. The normal operation of this system is coordinated by image recognition, automatic control, and other technical means [1]. The core application of the video surveillance system is the detection of moving targets. The purpose is to automatically process the collected video information through the monitoring equipment, obtain the moving objects in the video, and then carry out a series of deeper analysis, such as anomaly detection, target tracking, and behavior recognition. Corresponding to these moving objects in the video are static objects, which constitute the static background image of the video. However, people often pay more attention to the dynamic objects, and conduct a lot of research on the subsequent analysis of dynamic targets [2] but ignore the complete extraction of the static background image. To separate dynamic objects from these static backgrounds, background subtraction [3] is the most commonly used method. The moving foreground target can be obtained by subtracting the background image from the current frame image. Among them, the extraction of the video background image is the most critical step. Whether the background image in the video can be obtained accurately will directly affect the result of foreground target detection. Therefore, the extraction of the video background image is very important for foreground target detection and is a meaningful research direction. Figure 1 shows the video image coding transmission framework.Figure 1
Video image coding transmission framework.


## 2. Literature Review

Pretto and others proposed an image reconstruction algorithm based on image data set. The algorithm attempts to find similar image blocks from the image data set by using feature matching and reconstruct the original image by using similar image blocks. Although the reconstruction effect obtained by this algorithm is not ideal, the method of using local features to find the correlation between image blocks is worthy of reference [4]. Cao and others proposed a cloud based distributed coding algorithm. The algorithm uses the thumbnail of the image to be coded to search for similar images in the cloud dataset and uses the super-resolution reconstruction method similar to the cloud dataset to obtain the preliminary reconstructed image as the edge information of the distributed source coding and then iteratively optimizes between the final reconstruction and the edge information through the bit plane based checker coding to improve the reconstruction quality. This algorithm can obtain good color reconstruction and geometric reconstruction, but the iterative reconstruction process using edge information has high complexity [5]. Wang and others proposed a region prediction coding algorithm based on cloud storage image. Firstly, the reference image is divided into multiple planar regions according to the superpixel and local feature matching results; then, the geometric and illumination differences are compensated in the matching area between the reference image and the target image; finally, multiple references are generated from the estimated compensation model and organized into pseudosequences, so that the pseudosequences containing the input image can be differentially encoded by classical video coding tools to improve the rate distortion performance of the encoded input image [6]. In the process of coarse quantization of the original image based on the a priori quantization of the fine-grained image stored in the JPEG bin. Bin and others proposed a priori quantization of the image based on the a priori quantization of the sparse image. This algorithm can realize high-quality image reconstruction while saving the upload code stream [7].The application of the fault-tolerant coding method of digital multimedia video image has been widely concerned by the industry. Xc and others proposed a fault-tolerant coding method based on human visual characteristics. This method ignores that the original digital multimedia video image contains more visual redundancy, which seriously affects the coding/decoding speed and reconstruction quality of digital multimedia video image [8]. The histogram based fault-tolerant coding method studied by Wang and others makes the energy concentration effect worse when quantizing and coding the digital multimedia video image and finally leads to the obvious decline of the reconstructed image quality [9].Aiming at the defects and shortcomings of these methods, a fault-tolerant coding method based on wavelet transform and sparse decomposition is proposed. The experimental results fully prove that the proposed method can improve the quality of digital multimedia video image reconstruction and ensure the speed of image coding and decoding.


## 3. Research Methods

### 3.1. Fault Tolerant Digital Video Transmission System

The three parameters of spatial contrast sensitivity function, brightness adaptive factor, and contrast masking factor of the digital multimedia video image are calculated, respectively. According to the product of the three parameters, the transform domain JND model of the digital multimedia video image is established, and the model is used for the quantization and coding of multimedia video multiresolution image [10, 11].The JND model of digital multimedia video image based on the DCT transform domain can be described as the product of digital multimedia video image spatial contrast sensitivity function, image brightness adaptive factor, and image contrast masking factor. Its calculation formula is as follows:(1)JDCTi,j=JCSFi,j×Alum×Fcontrasti,j,where Alum represents the brightness adaptive weighting factor of the image; Fcontrasti,j represents the contrast masking weighting factor of the image; JCSFi,j represents the spatial contrast sensitivity function of a digital multimedia video image, which reflects the basic minimum detection error JND threshold of DCT transform domain coefficient with index number i,j, i,j in the image, and its expression is as follows [12]:(2)JCSFi,j=sϑiϑj×expcωij/a+bωijτ+1−τcosϑij.Here,s represents the collective effect of digital multimedia video images; ϑi and ϑj represent the normalization coefficient of DCT change domain, respectively; ωij represents the spatial frequency of the coefficient of image DCT variation domain; τ+1−τcosϑij represents the tilt effect of human vision; ϑij B represents the direction angle of the corresponding DCT coefficient of the digital multimedia video image; a and b represent the translation factor and expansion factor of the image spatial transformation domain, respectively.Alum in equation (1) shows that the brightness adaptive masking effect makes the sensitivity of human vision to different brightness areas in digital multimedia video images different. The value of Alum is related to the average brightness value I¯ of the local area of the image. The specific expression is as follows [13]:(3)Alum=1+60−I¯150,I¯≤60,1,60<I¯<170,1+I¯−170425,I¯≥60.Considering the masking effect of the digital multimedia video image subband coefficient itself, the calculation expression ofFcontrasti,j is as follows:(4)Fcontrasti,j=ψ,i2+j2≤16and in the smooth area or edge area,ψ×min4,max1,Ci,jJCSFi,j×Alum0.36,others,where ψ represents the weighting of digital multimedia video image; Ci,j represents the texture region masking effect of a digital multimedia video image.When calculating the contrast masking weighting factor of the digital multimedia video image, first divide the digital multimedia video image into three regions, namely, smooth region, texture region, and edge region. Weight different regions of the image according to formula (4).According to the above-given steps, the transform domainJND model of the digital multimedia video image can be established [14].The digital multimedia video image is divided into several subimages, and the digital multimedia video image is quantized and encoded by the following formula (5). The specific expression is as follows:(5)Mμ,κ=JDCTi,jCμ,κZμ,κ,where Cμ,κ represents the coefficient matrix of multimedia video image; Zμ,κ represents the transformation matrix of d-coefficient selection and quantization of multimedia video image.To sum up, it is the fault-tolerant coding principle of a multimedia video image. According to this principle, the coding and transmission of the digital video image is completed [15].


### 3.2. Fault Tolerant Coding Transmission Method of Digital Multimedia Video Image

#### 3.2.1. Wavelet Transform and Vector Quantization of Multi-Resolution Image

The speed of fault-tolerant coding of digital multimedia video image and the quality of reconstructed image seriously restrict the effect of fault-tolerant coding of the multimedia video image. The method of wavelet transform is used to decompose and reconstruct the digital multimedia video image; vector quantization is carried out on the video image after wavelet decomposition. The mean square deviation method and similar scalar optimal quantization method are used to select and calculate the multimedia video image vector, respectively, and each reconstruction vector is deduced to realize the preliminary image coding and overcome the block effect [16, 17].The wavelet transform of digital multimedia video image solves the disadvantage that the DCT transform used in the current compression method is limited by the network bandwidth. The multiresolution transformation characteristics of wavelet transform enable the digital multimedia video image to make full use of the visual characteristics of human eyes and retain the detail information of the original image under various resolutions. Suppose thath and g, respectively, represent the low-pass filter and high pass filter of the same wavelet base corresponding to the multimedia video image; Cj−1 represents the original digital multimedia video image; Cj represents the brightness value of the digital multimedia video image after wavelet transform; dj1, dj2, and dj3 represent the edge subimage of digital multimedia video image after wavelet transform [18]. Then, the wavelet decomposition calculation formula of a digital multimedia video image is as follows (6)∼(9):(6)Cjn,m=12∑k,l∈ZCj−1k,lhk−2nhl−2m,(7)dj1n,m=12∑k,l∈ZCj−1k,lgk−2nhl−2m,(8)dj2n,m=12∑k,l∈ZCj−1k,lhk−2ngl−2m,(9)dj3n,m=12∑k,l∈ZCj−1k,lgk−2ngl−2m,where k represents the dimension of digital multimedia video image; n represents the number of vectors of multimedia video images; l represents the wavelet coefficient of multimedia video image; m represents the continuous wavelet generating function of a multimedia video image.Then, the wavelet reconstruction calculation expression of the digital multimedia video image is as follows:(10)Ck−1n,m=12∑k,l∈ZCj−1k,lhk−2nhl−2m+∑k,l∈ZCj−1k,lhk−2ngl−2m+∑k,l∈ZCj−1k,lgk−2nhl−2m+dj3n,m=12∑k,l∈ZCj−1k,lgk−2ngl−2m.According to the above-given calculation, the wavelet decomposition of a digital multimedia video image is realized, and the original signal of a multiresolution image is decomposed into four subbands, including an approximation signal represented asLL (the low-frequency component of a multiresolution image, including the main energy of image) and three detail signals represented as LH, HL, and HH (the high-frequency component of a multiresolution image, including the detail information of image) [19].The original digital multimedia video image is divided into several subgraphs ofm×n each subgraph of the multiresolution image can be rewritten into k=m×n -dimensional vector form and its expression is as follows:(11)fifi1,fi2,…,fik′.Thek dimensional space of digital multimedia video image is decomposed into subspaces, that is, Dll=0,1,…,L−1. Assuming fi∈Dl, a known and determined vector γl in Dl is used to replace fi, that is, any image vector in Dl is quantized. The mean square deviation method is used to select the digital multimedia video image vector γl, and the expression is as follows:(12)e2=∑l=0L−1∫Dlfi−γl′fi−γlpfi=min,where pfi is the probability density function of fi.Using the scalar like optimal quantization method to calculate the digital multimedia video image vectorγl can be obtained.(13)γl=∫Dlfipfi∫Dlpfi.For any digital multimedia video image, each reconstruction vectorγl can be deduced according to formula (13), and all γl obtained can be calculated to generate a table a [20]. The completion of vector quantization of digital multimedia video image can overcome the block effect of other coding methods and enable the digital multimedia video image to obtain high image quality at a low bit rate, realize the preliminary image coding, and lay a good data foundation for the subsequent implementation of image fault-tolerant coding.


#### 3.2.2. Multiresolution Image Coding Based on Wavelet Transform and Vector Quantization

Wavelet transform can well remove the correlation between pixels and has the characteristics of multiresolution analysis. Many researchers have studied the image coding algorithm in the wavelet transform domain and achieved a good compression effect. The image coding and decoding framework based on wavelet transform is shown in Figure2.Figure 2
Image CODEC framework based on wavelet transform.Based on the preliminary coding of digital multimedia video images, the over complete database of digital multimedia video images is constructed and normalized. The digital multimedia video image is thinly decomposed by asymmetric atoms to obtain a linear representation of the image; using a few base atoms as the main components of the image, the asymmetric atoms of the image are transformed by rotation, translation, and expansion, and the type parameters are introduced to construct the atom library [21]. According to the above operations, we can understand and master the distribution range and distribution law of digital multimedia video image pixels and realize image fault-tolerant coding.Suppose that the digital multimedia video image is represented asX and the image size is represented as M1 × M2, where M1 and M2 represent the length and width of the digital multimedia video image, respectively. If the multiresolution image can be decomposed on a set of complete orthogonal bases, then M1 × M2 is the number of these bases, and the over a complete library of the multiresolution image is the set composed of these bases.SupposeD=gr,r∈Γ represents the sparse decomposition over a complete library of digital multimedia video images, and gr represents the base atom defined by the parameter group r of the over a complete library. After normalization, the following formula can be obtained:(14)gr=1.Assuming that the image sparse decomposition exceeds the number of base atoms in the complete libraryD, then(15)P≫M1×M2.According to the sparse decomposition of the multimedia video multiresolution image, a linear representation of the multiresolution image is obtained as follows:(16)X=∑k=0∞RkX,grkgrk,where RkX,grk represents the component of base atom grk corresponding to a digital multimedia video image. A few base atoms are used to represent the main components of a digital multimedia video image, and the calculation expression is as follows:(17)X≈∑k=0n−1RkX,grkgrk.Fault tolerant coding of a digital multimedia video image, that is, quantization and coding of the data of equation (17). The asymmetric atom is used for sparse decomposition of a digital multimedia video image, and its basic expression is as follows:(18)gx,y=1πε−x2+y2+23π4x2−2ε−x2+y2,where in, the first item of equation (18) represents that the low-frequency component of the digital multimedia video image is smooth in all directions; the second item represents the smoothing of the low-frequency component of the image in the same direction [22]. Which is mainly used to capture the edge information and contour information of multimedia video multiresolution image.The asymmetric atoms of the image are rotated, translated, and expanded, and the type parameters are introduced to construct the atomic Library of multimedia multiresolution images. The calculation expression is as follows:(19)D=gr,r∈Γ=1sxsygφx−usx,x−usy,where r=φ,u,v,sx,sy; φ represents the rotation of multimedia video multiresolution image; u and υ represent the translation of the image in the x direction and y direction, respectively; sx and sy represent the scale of the multiresolution image in x direction and y direction, respectively [23].According to equations (14)–(19), the final data obtained by sparse decomposition of a digital multimedia video image is as follows:(20)RkX,grk,φk,u,v,sx,sy.Analyze the data calculated by formula (20), understand, and master the distribution range and distribution law of digital multimedia video image pixels, so as to realize fault-tolerant coding [24].


## 3.1. Fault Tolerant Digital Video Transmission System

The three parameters of spatial contrast sensitivity function, brightness adaptive factor, and contrast masking factor of the digital multimedia video image are calculated, respectively. According to the product of the three parameters, the transform domain JND model of the digital multimedia video image is established, and the model is used for the quantization and coding of multimedia video multiresolution image [10, 11].The JND model of digital multimedia video image based on the DCT transform domain can be described as the product of digital multimedia video image spatial contrast sensitivity function, image brightness adaptive factor, and image contrast masking factor. Its calculation formula is as follows:(1)JDCTi,j=JCSFi,j×Alum×Fcontrasti,j,where Alum represents the brightness adaptive weighting factor of the image; Fcontrasti,j represents the contrast masking weighting factor of the image; JCSFi,j represents the spatial contrast sensitivity function of a digital multimedia video image, which reflects the basic minimum detection error JND threshold of DCT transform domain coefficient with index number i,j, i,j in the image, and its expression is as follows [12]:(2)JCSFi,j=sϑiϑj×expcωij/a+bωijτ+1−τcosϑij.Here,s represents the collective effect of digital multimedia video images; ϑi and ϑj represent the normalization coefficient of DCT change domain, respectively; ωij represents the spatial frequency of the coefficient of image DCT variation domain; τ+1−τcosϑij represents the tilt effect of human vision; ϑij B represents the direction angle of the corresponding DCT coefficient of the digital multimedia video image; a and b represent the translation factor and expansion factor of the image spatial transformation domain, respectively.Alum in equation (1) shows that the brightness adaptive masking effect makes the sensitivity of human vision to different brightness areas in digital multimedia video images different. The value of Alum is related to the average brightness value I¯ of the local area of the image. The specific expression is as follows [13]:(3)Alum=1+60−I¯150,I¯≤60,1,60<I¯<170,1+I¯−170425,I¯≥60.Considering the masking effect of the digital multimedia video image subband coefficient itself, the calculation expression ofFcontrasti,j is as follows:(4)Fcontrasti,j=ψ,i2+j2≤16and in the smooth area or edge area,ψ×min4,max1,Ci,jJCSFi,j×Alum0.36,others,where ψ represents the weighting of digital multimedia video image; Ci,j represents the texture region masking effect of a digital multimedia video image.When calculating the contrast masking weighting factor of the digital multimedia video image, first divide the digital multimedia video image into three regions, namely, smooth region, texture region, and edge region. Weight different regions of the image according to formula (4).According to the above-given steps, the transform domainJND model of the digital multimedia video image can be established [14].The digital multimedia video image is divided into several subimages, and the digital multimedia video image is quantized and encoded by the following formula (5). The specific expression is as follows:(5)Mμ,κ=JDCTi,jCμ,κZμ,κ,where Cμ,κ represents the coefficient matrix of multimedia video image; Zμ,κ represents the transformation matrix of d-coefficient selection and quantization of multimedia video image.To sum up, it is the fault-tolerant coding principle of a multimedia video image. According to this principle, the coding and transmission of the digital video image is completed [15].


## 3.2. Fault Tolerant Coding Transmission Method of Digital Multimedia Video Image

### 3.2.1. Wavelet Transform and Vector Quantization of Multi-Resolution Image

The speed of fault-tolerant coding of digital multimedia video image and the quality of reconstructed image seriously restrict the effect of fault-tolerant coding of the multimedia video image. The method of wavelet transform is used to decompose and reconstruct the digital multimedia video image; vector quantization is carried out on the video image after wavelet decomposition. The mean square deviation method and similar scalar optimal quantization method are used to select and calculate the multimedia video image vector, respectively, and each reconstruction vector is deduced to realize the preliminary image coding and overcome the block effect [16, 17].The wavelet transform of digital multimedia video image solves the disadvantage that the DCT transform used in the current compression method is limited by the network bandwidth. The multiresolution transformation characteristics of wavelet transform enable the digital multimedia video image to make full use of the visual characteristics of human eyes and retain the detail information of the original image under various resolutions. Suppose thath and g, respectively, represent the low-pass filter and high pass filter of the same wavelet base corresponding to the multimedia video image; Cj−1 represents the original digital multimedia video image; Cj represents the brightness value of the digital multimedia video image after wavelet transform; dj1, dj2, and dj3 represent the edge subimage of digital multimedia video image after wavelet transform [18]. Then, the wavelet decomposition calculation formula of a digital multimedia video image is as follows (6)∼(9):(6)Cjn,m=12∑k,l∈ZCj−1k,lhk−2nhl−2m,(7)dj1n,m=12∑k,l∈ZCj−1k,lgk−2nhl−2m,(8)dj2n,m=12∑k,l∈ZCj−1k,lhk−2ngl−2m,(9)dj3n,m=12∑k,l∈ZCj−1k,lgk−2ngl−2m,where k represents the dimension of digital multimedia video image; n represents the number of vectors of multimedia video images; l represents the wavelet coefficient of multimedia video image; m represents the continuous wavelet generating function of a multimedia video image.Then, the wavelet reconstruction calculation expression of the digital multimedia video image is as follows:(10)Ck−1n,m=12∑k,l∈ZCj−1k,lhk−2nhl−2m+∑k,l∈ZCj−1k,lhk−2ngl−2m+∑k,l∈ZCj−1k,lgk−2nhl−2m+dj3n,m=12∑k,l∈ZCj−1k,lgk−2ngl−2m.According to the above-given calculation, the wavelet decomposition of a digital multimedia video image is realized, and the original signal of a multiresolution image is decomposed into four subbands, including an approximation signal represented asLL (the low-frequency component of a multiresolution image, including the main energy of image) and three detail signals represented as LH, HL, and HH (the high-frequency component of a multiresolution image, including the detail information of image) [19].The original digital multimedia video image is divided into several subgraphs ofm×n each subgraph of the multiresolution image can be rewritten into k=m×n -dimensional vector form and its expression is as follows:(11)fifi1,fi2,…,fik′.Thek dimensional space of digital multimedia video image is decomposed into subspaces, that is, Dll=0,1,…,L−1. Assuming fi∈Dl, a known and determined vector γl in Dl is used to replace fi, that is, any image vector in Dl is quantized. The mean square deviation method is used to select the digital multimedia video image vector γl, and the expression is as follows:(12)e2=∑l=0L−1∫Dlfi−γl′fi−γlpfi=min,where pfi is the probability density function of fi.Using the scalar like optimal quantization method to calculate the digital multimedia video image vectorγl can be obtained.(13)γl=∫Dlfipfi∫Dlpfi.For any digital multimedia video image, each reconstruction vectorγl can be deduced according to formula (13), and all γl obtained can be calculated to generate a table a [20]. The completion of vector quantization of digital multimedia video image can overcome the block effect of other coding methods and enable the digital multimedia video image to obtain high image quality at a low bit rate, realize the preliminary image coding, and lay a good data foundation for the subsequent implementation of image fault-tolerant coding.


### 3.2.2. Multiresolution Image Coding Based on Wavelet Transform and Vector Quantization

Wavelet transform can well remove the correlation between pixels and has the characteristics of multiresolution analysis. Many researchers have studied the image coding algorithm in the wavelet transform domain and achieved a good compression effect. The image coding and decoding framework based on wavelet transform is shown in Figure2.Figure 2
Image CODEC framework based on wavelet transform.Based on the preliminary coding of digital multimedia video images, the over complete database of digital multimedia video images is constructed and normalized. The digital multimedia video image is thinly decomposed by asymmetric atoms to obtain a linear representation of the image; using a few base atoms as the main components of the image, the asymmetric atoms of the image are transformed by rotation, translation, and expansion, and the type parameters are introduced to construct the atom library [21]. According to the above operations, we can understand and master the distribution range and distribution law of digital multimedia video image pixels and realize image fault-tolerant coding.Suppose that the digital multimedia video image is represented asX and the image size is represented as M1 × M2, where M1 and M2 represent the length and width of the digital multimedia video image, respectively. If the multiresolution image can be decomposed on a set of complete orthogonal bases, then M1 × M2 is the number of these bases, and the over a complete library of the multiresolution image is the set composed of these bases.SupposeD=gr,r∈Γ represents the sparse decomposition over a complete library of digital multimedia video images, and gr represents the base atom defined by the parameter group r of the over a complete library. After normalization, the following formula can be obtained:(14)gr=1.Assuming that the image sparse decomposition exceeds the number of base atoms in the complete libraryD, then(15)P≫M1×M2.According to the sparse decomposition of the multimedia video multiresolution image, a linear representation of the multiresolution image is obtained as follows:(16)X=∑k=0∞RkX,grkgrk,where RkX,grk represents the component of base atom grk corresponding to a digital multimedia video image. A few base atoms are used to represent the main components of a digital multimedia video image, and the calculation expression is as follows:(17)X≈∑k=0n−1RkX,grkgrk.Fault tolerant coding of a digital multimedia video image, that is, quantization and coding of the data of equation (17). The asymmetric atom is used for sparse decomposition of a digital multimedia video image, and its basic expression is as follows:(18)gx,y=1πε−x2+y2+23π4x2−2ε−x2+y2,where in, the first item of equation (18) represents that the low-frequency component of the digital multimedia video image is smooth in all directions; the second item represents the smoothing of the low-frequency component of the image in the same direction [22]. Which is mainly used to capture the edge information and contour information of multimedia video multiresolution image.The asymmetric atoms of the image are rotated, translated, and expanded, and the type parameters are introduced to construct the atomic Library of multimedia multiresolution images. The calculation expression is as follows:(19)D=gr,r∈Γ=1sxsygφx−usx,x−usy,where r=φ,u,v,sx,sy; φ represents the rotation of multimedia video multiresolution image; u and υ represent the translation of the image in the x direction and y direction, respectively; sx and sy represent the scale of the multiresolution image in x direction and y direction, respectively [23].According to equations (14)–(19), the final data obtained by sparse decomposition of a digital multimedia video image is as follows:(20)RkX,grk,φk,u,v,sx,sy.Analyze the data calculated by formula (20), understand, and master the distribution range and distribution law of digital multimedia video image pixels, so as to realize fault-tolerant coding [24].


## 3.2.1. Wavelet Transform and Vector Quantization of Multi-Resolution Image

The speed of fault-tolerant coding of digital multimedia video image and the quality of reconstructed image seriously restrict the effect of fault-tolerant coding of the multimedia video image. The method of wavelet transform is used to decompose and reconstruct the digital multimedia video image; vector quantization is carried out on the video image after wavelet decomposition. The mean square deviation method and similar scalar optimal quantization method are used to select and calculate the multimedia video image vector, respectively, and each reconstruction vector is deduced to realize the preliminary image coding and overcome the block effect [16, 17].The wavelet transform of digital multimedia video image solves the disadvantage that the DCT transform used in the current compression method is limited by the network bandwidth. The multiresolution transformation characteristics of wavelet transform enable the digital multimedia video image to make full use of the visual characteristics of human eyes and retain the detail information of the original image under various resolutions. Suppose thath and g, respectively, represent the low-pass filter and high pass filter of the same wavelet base corresponding to the multimedia video image; Cj−1 represents the original digital multimedia video image; Cj represents the brightness value of the digital multimedia video image after wavelet transform; dj1, dj2, and dj3 represent the edge subimage of digital multimedia video image after wavelet transform [18]. Then, the wavelet decomposition calculation formula of a digital multimedia video image is as follows (6)∼(9):(6)Cjn,m=12∑k,l∈ZCj−1k,lhk−2nhl−2m,(7)dj1n,m=12∑k,l∈ZCj−1k,lgk−2nhl−2m,(8)dj2n,m=12∑k,l∈ZCj−1k,lhk−2ngl−2m,(9)dj3n,m=12∑k,l∈ZCj−1k,lgk−2ngl−2m,where k represents the dimension of digital multimedia video image; n represents the number of vectors of multimedia video images; l represents the wavelet coefficient of multimedia video image; m represents the continuous wavelet generating function of a multimedia video image.Then, the wavelet reconstruction calculation expression of the digital multimedia video image is as follows:(10)Ck−1n,m=12∑k,l∈ZCj−1k,lhk−2nhl−2m+∑k,l∈ZCj−1k,lhk−2ngl−2m+∑k,l∈ZCj−1k,lgk−2nhl−2m+dj3n,m=12∑k,l∈ZCj−1k,lgk−2ngl−2m.According to the above-given calculation, the wavelet decomposition of a digital multimedia video image is realized, and the original signal of a multiresolution image is decomposed into four subbands, including an approximation signal represented asLL (the low-frequency component of a multiresolution image, including the main energy of image) and three detail signals represented as LH, HL, and HH (the high-frequency component of a multiresolution image, including the detail information of image) [19].The original digital multimedia video image is divided into several subgraphs ofm×n each subgraph of the multiresolution image can be rewritten into k=m×n -dimensional vector form and its expression is as follows:(11)fifi1,fi2,…,fik′.Thek dimensional space of digital multimedia video image is decomposed into subspaces, that is, Dll=0,1,…,L−1. Assuming fi∈Dl, a known and determined vector γl in Dl is used to replace fi, that is, any image vector in Dl is quantized. The mean square deviation method is used to select the digital multimedia video image vector γl, and the expression is as follows:(12)e2=∑l=0L−1∫Dlfi−γl′fi−γlpfi=min,where pfi is the probability density function of fi.Using the scalar like optimal quantization method to calculate the digital multimedia video image vectorγl can be obtained.(13)γl=∫Dlfipfi∫Dlpfi.For any digital multimedia video image, each reconstruction vectorγl can be deduced according to formula (13), and all γl obtained can be calculated to generate a table a [20]. The completion of vector quantization of digital multimedia video image can overcome the block effect of other coding methods and enable the digital multimedia video image to obtain high image quality at a low bit rate, realize the preliminary image coding, and lay a good data foundation for the subsequent implementation of image fault-tolerant coding.


## 3.2.2. Multiresolution Image Coding Based on Wavelet Transform and Vector Quantization

Wavelet transform can well remove the correlation between pixels and has the characteristics of multiresolution analysis. Many researchers have studied the image coding algorithm in the wavelet transform domain and achieved a good compression effect. The image coding and decoding framework based on wavelet transform is shown in Figure2.Figure 2
Image CODEC framework based on wavelet transform.Based on the preliminary coding of digital multimedia video images, the over complete database of digital multimedia video images is constructed and normalized. The digital multimedia video image is thinly decomposed by asymmetric atoms to obtain a linear representation of the image; using a few base atoms as the main components of the image, the asymmetric atoms of the image are transformed by rotation, translation, and expansion, and the type parameters are introduced to construct the atom library [21]. According to the above operations, we can understand and master the distribution range and distribution law of digital multimedia video image pixels and realize image fault-tolerant coding.Suppose that the digital multimedia video image is represented asX and the image size is represented as M1 × M2, where M1 and M2 represent the length and width of the digital multimedia video image, respectively. If the multiresolution image can be decomposed on a set of complete orthogonal bases, then M1 × M2 is the number of these bases, and the over a complete library of the multiresolution image is the set composed of these bases.SupposeD=gr,r∈Γ represents the sparse decomposition over a complete library of digital multimedia video images, and gr represents the base atom defined by the parameter group r of the over a complete library. After normalization, the following formula can be obtained:(14)gr=1.Assuming that the image sparse decomposition exceeds the number of base atoms in the complete libraryD, then(15)P≫M1×M2.According to the sparse decomposition of the multimedia video multiresolution image, a linear representation of the multiresolution image is obtained as follows:(16)X=∑k=0∞RkX,grkgrk,where RkX,grk represents the component of base atom grk corresponding to a digital multimedia video image. A few base atoms are used to represent the main components of a digital multimedia video image, and the calculation expression is as follows:(17)X≈∑k=0n−1RkX,grkgrk.Fault tolerant coding of a digital multimedia video image, that is, quantization and coding of the data of equation (17). The asymmetric atom is used for sparse decomposition of a digital multimedia video image, and its basic expression is as follows:(18)gx,y=1πε−x2+y2+23π4x2−2ε−x2+y2,where in, the first item of equation (18) represents that the low-frequency component of the digital multimedia video image is smooth in all directions; the second item represents the smoothing of the low-frequency component of the image in the same direction [22]. Which is mainly used to capture the edge information and contour information of multimedia video multiresolution image.The asymmetric atoms of the image are rotated, translated, and expanded, and the type parameters are introduced to construct the atomic Library of multimedia multiresolution images. The calculation expression is as follows:(19)D=gr,r∈Γ=1sxsygφx−usx,x−usy,where r=φ,u,v,sx,sy; φ represents the rotation of multimedia video multiresolution image; u and υ represent the translation of the image in the x direction and y direction, respectively; sx and sy represent the scale of the multiresolution image in x direction and y direction, respectively [23].According to equations (14)–(19), the final data obtained by sparse decomposition of a digital multimedia video image is as follows:(20)RkX,grk,φk,u,v,sx,sy.Analyze the data calculated by formula (20), understand, and master the distribution range and distribution law of digital multimedia video image pixels, so as to realize fault-tolerant coding [24].


## 4. Result Analysis

Simulation test environment: the CPU processor is Intel Pentium E2200 3.2 GHz, the running memory is 1024 MB, and the operating system is Windows7. The simulation test tool uses MATLAB 7.11.0 to simulate a random digital multimedia video image with the size of 512 × 512. Peak signal-to-noise ratio (PSNR) and fault-tolerant coding speed are used as the coding performance metrics of this method and other methods. The test results are shown in Table1 and Figure 3.Table 1
Performance test results of other methods.
NumberCRPSNR/dBCodingDecoding58.141.2016.4915.24107.6638.1559.8857.66157.1546.72118.24103.87Figure 3
Performance test results of this method. (a) CR. (b) PSNR/dB. (c) Coding. (d) Decoding.
(a)(b)(c)(d)As can be seen from Table1 and Figure 3, Number represents the number of iterations of digital multimedia video image fault-tolerant coding; CR represents the compression ratio of digital multimedia video image; Coding represents the encoding time of digital multimedia video image; Decoding represents the decoding time of digital multimedia video image.By analyzing the simulation test data in Table1 and Figure 3 above, it can be seen that compared with other methods, the method proposed in this paper not only improves the reconstruction quality of digital multimedia video image but also greatly shortens the encoding and decoding time of digital multimedia video image. When the number of iterations is 15, the CR index is the same, PSNR increases by 8.7%, coding is 23.7% faster and decoding is 15% faster.


## 5. Conclusion

At present, most of the widely used fault-tolerant coding methods realize fault-tolerant coding at the expense of image quality or seriously restrict the coding speed even if it does not affect the image quality. The fault-tolerant coding method based on wavelet transform and sparse decomposition proposed in this paper not only improves the fault-tolerant coding speed of digital multimedia video image but also significantly improves the quality of image reconstruction. The simulation results show that compared with other methods, the performance of the proposed method has obvious advantages and broad application prospects. Based on the research of this paper, we can try to combine more different methods, so as to further strengthen and improve the transmission quality of digital multimedia video images.



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