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Let I be an image defined in image domain ��. The primary goal of Retinex theory is to decompose I into the reflectance image, R, and the illumination image, L, as shown in Figure 5, such that, at each point in the image domain [25]: I=R?L (2) and following [14,17], we may further assume selleckchem that: L��I>0 Figure 5 Schematicdiagram for Retinex. We first convert Equation (2) into the logarithmic domain: i=log(I),?l=log(L),?r=log(R) so that: i=l+r Based on our new assumption, the illumination image may contain non-smooth parts. Weuse a total variation like regularizer near non-smooth parts and a Tikhonov like regularizer for smooth parts. We minimize the objective function as follows: E(l)=�Ҧ�|?l??i|2dx+�ˡҦ�|?l|p(x)dx (3) where CHIR-99021 chemical structure �� is a positive number, and p(x)=1+11+w|?d|2, where d is the ideal illumination image, discussed in Section 2.4.2. The first fidelity term on the right side of model (3) measures the similarity of the gradient between the illumination and the original image, and the second is the regularization term. Clearly, p��1 near the edges of d where the gradient is large, and so the regularizer is similar to a TV regularizer which can preserve edges; p��2 in the homogeneous regions where the gradient is small, and here the regularizer is similar to a Tikhonov like regularizer, which is superior to total variation. In other regions, the penalty is adjusted by p(x). The classical Retinex algorithm uses a Gaussian filter, equivalent to a Tikhonov regularizer, to obtain the illumination image. However, a Gaussian filter smears edges, which is the main cause of halo artifacts [26]. Using the adaptive TV like regularizer for the high contrast edges in the image, our model not only prevents halo artifacts but also extracts Fleroxacin the edges of non-uniform illumination from the image. 2.3. Solution Existence Let us recall some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces, following [24,27]. Definition? (variable exponent spaces): Let �� be a bounded open set with Lipchitz boundary and p(x):����[1,+��) be a measurable function, with the family of all measurable functions on �� being P(��). We define a functional, which is also called modular: Qp(x)(u)=�Ҧ�|u|p(x)dx and a norm: ||u||p(x)=inf��>0:Qp(x)(u/��)��1 Then the variable exponent Lebesgue and Sobolev spaces are, respectively: Lp(x)(��)=u:����R and: W1,p(x)(��)=?u��Lp(x)(��),?u��Lp(x)(��) With the norm ||u||1,p(x)=||u||p(x)+||?u||p(x), W1,p(x)(��) becomes a Banach space. Lemma 1.? (relationship between modular and norm [27]): Let Qp(x) be a modular on X and u��X, then ||u||p(x)��Qp(x)(u)+1. Lemma 2.? (embedding theorem [24]): Let p(x),q(x)��P(��) , and p(x)��q(x) for a.e. x�ʦ�. Then Lq(x)(��) is continuously embedded in Lp(x)(��). Lemma 3.? (convexity [27]): Let F(?l,x)=|?l|p(x) , with p(x)=1+11+w|?d|2 as in model (3).