The dictionary based image sparsity is used to construct a denoising variational model, and a weight factor is designed to suppress the interference of the impact noise in the data fidelity term in the model. The initial denoising of mixed noise images is achieved through the idea of non-local mean. In the obtained image, the masking matrix is constructed to exclude the impact noise points and obtain non-local similarity prior knowledge. The non-local similarity prior and sparse prior are fused into the regular terms of the variational model, and a mixed denoising model is constructed to obtain the denoised face image.

The general accuracy of input face images is high and there is a lot of redundant information. In order to avoid the complex structure of genetic neural network, image compression must be performed before the genetic neural network is input [10]. Let *f* (*x*) be used for interpolation and *h*(*x*) for interpolation kernel, then the interpolation Eq. is:

$$\begin{array}{r}f(x)=\sum _{i=0}^{K-1}{C}_{i}h(x-{x}_{i})\end{array}$$(1)where, *C *_{i} represents the weight coefficient. The face recognition method based on the GA-BP neural network algorithm uses cubic function, and the size of the interpolated neighborhood is 4×4. Thus, the value of the output pixel value is the weighted average of the effective points contained in the 4×4 matrix.

*x* represents the compressed face image; *x*_{i}_{,j} represents the pixel location located at the position (*i*, *j*); *y* represents the image after the face image *x* is contaminated by noise, and the expression of the mixed noise model is as follows:

$${y}_{i,j}=\left\{\begin{array}{l}{d}_{min}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{i}\mathrm{s}s/2\\ {d}_{max}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{i}\mathrm{s}s/2\\ {d}_{i,j}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{i}\mathrm{s}r(1-s)\\ {x}_{i,j}+{v}_{i,j}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{i}\mathrm{s}(1-r)(1-s)\end{array}\right.$$(2)where, *v*_{i}_{,j} is the Gauss noise value which obeys the zero mean distribution and the variance is *σ*.

Set [*d*_{min}, *d*_{max} ] be the range of the gray value of the noise image *y*. When the face image is contaminated with salt and pepper noise, *s*(0 ≤ *s* ≤ 1) is the ratio of the noise of salt and pepper. The value of the noise point is only two extreme cases: *d*_{min} or *d*_{max}, and their probability is *s* /2. For random impulse noise, the noise ratio is *r*(0 ≤ *r* ≤ 1), the noise point value is *d*_{i}_{,j} (*d*_{min} < *d*_{i}_{,j} < *d*_{max}), and the probability of *y*_{i}_{,j} = *d*_{i}_{,j} is *r*(1 − *s*). The pixel value that is only contaminated by Gauss noise is *y*_{i}_{,j} = *x*_{i}_{,j} + *v*_{i}_{,j} and the probability is (1 − *r*)(1 − *s*).

Let *x*_{i} ∈ R_{i} represent an image neighborhood block with the size of$\sqrt{n}\times \sqrt{n}$and *R*_{i} be a matrix vector. Through it, one image block *x*_{i} in image *x* can be removed. According to the theory of sparse representation, dictionary *Φ* = [*Φ*_{1}, *Φ*_{2}, · · · , *Φ*_{n}] can be used to make sparse coding of *x*_{i}, then *x*_{i} can be expressed as:

$$\begin{array}{r}{x}_{i}=\mathrm{\Phi}\cdot {\alpha}_{i},\end{array}$$(3)where, _{i} is a sparse coding vector. The encoded image *x* can be expressed as:

$$x=\mathrm{\Phi}\cdot \alpha $$(4)where, *α* represents a sparse coding vector. The choice of dictionary has a great influence on the sparse encoding and reconstruction of signals [11]. In the GA-BP neural network algorithm, the dictionary must be learned in advance through uncontaminated natural face images. By learning from natural face images, a PAC dictionary is used for sparse representation. PCA dictionary is self-adaptive. Principal components are extracted by principal component analysis (PCA) to update the atoms in the dictionary [12]. For image *y* contaminated by Gauss noise, coding model *α*_{new} can be expressed as:

$$\begin{array}{r}{\alpha}_{new}=\underset{\alpha}{\mathrm{arg}min}||y-\mathrm{\Phi}\cdot \alpha |{|}_{2}^{2}+R(\alpha ),\end{array}$$(5)where, *R*(*α*) is a regular term, which is to describe some properties of the solution to limit the solution space, so that the solution of the problem has favorite properties and thus a stable solution is obtained. According to the deterministic regularized theory, the maximum posterior solution of the coding vector obtained by Gauss noise is obtained [13]. For the face images contaminated by mixed noise, the distribution of noise is very different from that of Gauss noise. Under the influence of impact noise, the residual error *y* −*Φα* of data fitting for mixed noise is more irregular than that of Gauss noise. Therefore, in Eq. (5) *l*_{2} norm is used to represent that the data fitting residuals is not applicable to the restoration of mixed noise contaminated images. If the data fidelity term is changed to make the distribution of the residual difference more similar to the distribution of Gauss noise, the *l*_{2} norm can still be used to represent the coded residual under mixed noise, making the removal of the mixed noise easier to handle. By weighting data fidelity items, the distribution is more regular. So, let:

$$\begin{array}{r}e=[{e}_{1},{e}_{2},\cdots ,{e}_{N}]=y-\mathrm{\Phi}\alpha ,\end{array}$$(6)where, the residual ${e}_{i}=y-\mathrm{\Phi}\alpha .min\sum _{i=1}^{N}f({e}_{i})$is used to replace *e*_{i}. Function *f* controls the distribution of every residual, so that the residual distribution is more consistent with the residual distribution under Gauss noise. *f* should meet the following nature:

$$f(e)\ge 0\phantom{\rule{thickmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thickmathspace}{0ex}}f(0)=0$$$$f({e}_{i})\ge f({e}_{j}),|{e}_{i}|>|{e}_{j}|$$$$f(e)=f(-e)$$(7)The residual distribution in mixed noise will have heavy tailed distribution. The heavy-tailed distribution is similar to the normal probability distribution, but its tail is thicker than the tail of the normal distribution, with the characteristics of peak and heavy-tailed [14]. For each residual error, the weighted residuals are:

$$\begin{array}{r}{e}_{i}^{w}={w}_{i}^{1/2}{e}_{i},\end{array}$$(8)where, *w*_{i} is a diagonal element. In the mixed denoising, the residual can be divided into two categories: for the noise contaminated by Gauss noise, the residual error basically obeys the Gauss distribution and can remain unchanged, that is, the weight is approximately 1; Residues at impact noise points should be weighted to reduce the heavy tail distribution [15]. From the above analysis, *f* (*e*_{i}) can be set to $f({e}_{i})=({w}_{i}^{1/2}{e}_{i}{)}^{2},$and the improved mixed noise denoising model is:

$$\begin{array}{r}{\alpha}_{new}=\underset{\alpha}{\mathrm{arg}min}||{w}^{1-2}(y-\mathrm{\Phi}\alpha )|{|}_{2}^{2}+R(\alpha ),\end{array}$$(9)where, *w* is diagonal weighting matrix and diagonal element *w*_{ii} = *w*_{i}. The pixels contaminated by impact noise should have a smaller weight, which can reduce the influence of impact noise on coding. For pixels that are not contaminated by impact noise, the weight value should be close to 1. Because in the sparse representation of the impact noise points, there will be large coding residuals, which is set according to the encoding residual *e*_{i} to the weight *w*_{ii}, and the size of *w*_{ii} and *e*_{i} is inverse proportional. Set *w*_{ii} ∈ [0, 1], the calculation formula of *w*_{ii} is as follows:

$$\begin{array}{r}{w}_{ii}=\mathrm{exp}(-b{e}_{i}^{2}),\end{array}$$(10)where, *b* is a normal number, which can control the decay rate of *w*_{ii}. In order to make the denoising model (9) more effective for mixed noise removal, it is necessary to apply prior knowledge of the face image to the design of regular term *R*(*α*). The two prior knowledge of local sparsity and non local self-similarity are fused into a regular term of a variational model, and a more accurate sparse representation coefficient will be obtained for the construction of the denoising model.

When the face image contains the mixed noise composed of Gauss noise and impact noise, it can cause interference to the non local similarity solution in the face image [16]. The face recognition method based on GA-BP neural network firstly preprocesses the noise image *y*, and preliminarily removes the impulse noise in the face image. Using fuzzy weight non local average algorithm, the initial image *x*^{(0)} can be obtained by filtering the noise image. The fuzzy weight non local average algorithm uses the non local similarity of the face image, and the noise of the face image is removed by obtaining the fuzzy weighted non local mean value of the pixel points. It is more accurate to seek nonlocal similarity prior knowledge through *x*^{(0)}. The matrix *z* is constructed according to the initial image *x*^{(0)}:

$$z=||y-{x}^{(0)}||$$(11)A threshold value of 2*σ* is set to judge whether the pixel at the position (*i*, *j*) in noise image *y* is the impact noise point. If *z*(*i*, *j*) ≥ 2*σ*, it is determined that the pixel is the impact noise point, otherwise it is the Gauss noise point. An identification matrix *d*(*i*, *j*) is constructed by the matrix *z*:

$$d(i,j)=\left\{\begin{array}{l}1\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}z(i,j)<2\sigma \\ 0\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}z(i,j)\ge 2\sigma \end{array}\right.$$(12)When solving the non local similar priori of face image *x*^{(0)}, matrix *d*(*i*, *j*) is used as a mask matrix, and a new image matrix *x*^{(1)} is preliminarily excluded from the impact noise point in *x*^{(0)}, that is, the non local similarity prior to the face image in Gauss noise environment is obtained only. The result will be closer to the true information of the face image [17]. For the image block *x*_{i} in *x*^{(1)}, a similar block is looked for in the large enough window area centered on *i*. If the Euclidean distance between image blocks ${x}_{i}^{q}$and *x*_{i} does not exceed a predetermined threshold, then they are considered to be similar. If there are *L* image blocks similar to *x*_{i}, the weighted average of them can be obtained as following.

$$\begin{array}{r}{x}_{i}=\sum _{q=1}^{L}{b}_{i}^{q}{x}_{i}^{q},\end{array}$$(13)where, *q* represents the number of image blocks. *x*_{i} can be used to estimate image block, and the distance between weights ${b}_{i}^{q}$and *x*_{i} is inversely proportional to ${x}_{i}^{q}.$

$$\begin{array}{r}{b}_{i}^{q}=\mathrm{exp}(-||{x}_{i}-{x}_{i}^{q}|{|}_{2}^{2}/h)/\omega ,\end{array}$$(14)where, *h* is a pre-set scalar, and *⍵* is normalization factor. If the advanced learning dictionary *Φ* is used to encode the image block and its non local similar block, then *x*_{i} = *Φα*_{i}, and the non local similar block *x*_{i} = *Φα*_{i}, the coding coefficients ${\alpha}_{i}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{i}$are similar. The face recognition method based on GA-BP neural network algorithm takes $\sum _{i}||{\alpha}_{i}-{\mu}_{i}||$as a regular term to substitute into Eq. (9) then obtaining.

$$\begin{array}{r}{\alpha}_{new}=\underset{\alpha}{\mathrm{arg}min}||{w}^{1/2}(y-\mathrm{\Phi}\alpha )|{|}_{2}^{2}+||\alpha -\mu |{|}_{1},\end{array}$$(15)where, represents the coding coefficient. In the above model, the data fidelity term is weighted. The regular term combines the sparse prior and the non local similar priori, which makes the denoising model integrate more prior knowledge, and the denoising image will be closer to the real life image [18].

Under the mixed noise composed of Gaussian noise and impulse noise, the initial value of coding residue *e* can be set to *e*^{(0)} by the initial image *e*^{(0)}, and the calculation formula of *e*^{(0)} is as follows:

$${e}^{(0)}=y-{x}^{(0)}$$(16)The weighting matrix *w* is initialized by *e*^{(0)} through Eq. (10)

When *w* is determined, the algorithm model (15) becomes a sparse coding problem of *l*_{1} norm. Face recognition algorithm based on GA-BP neural network algorithm can solve the model by iterative reweighting algorithm. Let *V* be a diagonal matrix and initialize it as a unit matrix. At the *k* + 1th iteration, the element in *V* is updated to:

$$\begin{array}{r}{V}_{ii}^{(k+1)}=\mu /(({\alpha}_{i}^{(k)}-{\mu}_{i}{)}^{2}+{\epsilon}^{2}{)}^{1/2},\end{array}$$(17)where, *ε* is scalar, and the mixed denoising model *α*^{(k+1)} is obtained after update. The expression of *α*^{(k+1)} is as follows:

$${\alpha}^{(k+1)}=({\mathrm{\Phi}}^{T}w\mathrm{\Phi}+{V}_{ii}^{(k+1)}{)}^{-1}({\mathrm{\Phi}}^{T}wy-{\mathrm{\Phi}}^{T}w\mathrm{\Phi}\mu )+\mu $$(18)
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