12.2 Statistical Inference-Based Approaches

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where REDUCE( ) is an operator for smoothing and down-sampling, and it is actually chosen to be the pixel averaging function: REDUCE(I)(m, n) = 1 4

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I(2m + i, 2n + j)

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(12.8)

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In terms of the Gaussian pyramid, super-resolution is a function to obtain a highresolution image G0 (I) from an input low-resolution image Gk (I) where k > 0. In reference 1, it is assumed that Gk (I)(m, n), a pixel in the low-resolution image, is de ned as the addition of the weighted sum of the high-resolution pixels and i.i.d. Gaussian noise (m, n): Gk (I)(m, n) =

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W(m, n, p, q)G0 (I)(p, q) + (m, n).

(12.9)

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12.2.1.2 Estimation of a High-Resolution Image by the MAP Estimator and the Gradient Prior The high-resolution image G0 can be inferred for a given low-resolution image Gk by the MAP estimator in Eq. (12.4): G = arg max P(G0 |Gk ) = arg max P(Gk |G0 )P(G0 ). 0

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G0 G0

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(12.10)

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So, in the above equation, the MAP estimator can be applied if the two terms P(Gk |G0 ) and P(G0 ) are de ned. The rst term P(Gk |G0 ) can be solved by the assumption that (m, n) is i.i.d. Gaussian since P(Gk |G0 ) = P( (m, n)) in Eq. (12.9). Next, the prior term P(G0 ) is learned by a pyramid-based algorithm using a pyramids of feature vectors: Laplacian pyramids, the horizontal and vertical rst derivatives of the Gaussian pyramids, and the horizontal and vertical second derivatives of the Gaussian pyramids [1]. A feature vector Fk (Ti ) denotes a set of these pyramids for the training image Ti at the kth level. Here, if (m, n) is a pixel in the lth level of a pyramid, its parent at the l + 1th level is m , n . Thus, the parent structure vector of a pixel 2 2 (m, n) in the lth level is de ned as PSl (I)(m, n) = Fl (I)(m, n), . . . , FN (I) n m , N 1 2N 1 2 . (12.11)

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Finally, the optimal feature vector F0 (I)(m, n) of the unknown high-resolution )(m, n), where the training image Ti minimizes the image can be chosen to be F0 (Ti L2 norm error of the parent structure vectors:

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i = arg min PSk (I)

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m 2N 1

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PSk (Ti )

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n 2N 1

(12.12)

The high-resolution image G0 (m, n) can be easily obtained by F0 (I)(m, n).

12

Super-Resolution of Face Images

Two-Step Approach by Liu et al.

In references 9 and 19, it is assumed that a high-resolution face image h is a composition of a global face image hg and a local feature image hl : h = hg + hl . (12.13)

So, the two components are recovered respectively by a two-step approach: a global parametric model and a local nonparametric model. This assumption is based on the global and local constraints. First, the global constraint is that the result of face super-resolution must have common features of a human face for example, eyes, mouth, nose, symmetry, and so on. The rst step is for the global parametric modeling of face images based on the assumption that a global face image is generated by a Gaussian distribution learned by principal component analysis (PCA) [10]. Here, the PCA coef cients of the global face image is inferred by a MAP estimator introduced in Session 12.2. Next, the local constraint is that the result must have speci c characteristics of a face image with local features that make the face look different from other faces. Thus, the second step is for the local nonparametric modeling of face images and an optimal local feature image is inferred from the optimal global image by minimizing the energy of the Markov network. Finally, an output high-resolution image is obtained by the sum of the global and local images. In sum, the novelty of this approach is that by integrating both global and local models, both common feature and individual characteristics of faces are recovered respectively, so more reliable results can be yields. 12.2.2.1 Global Modeling: A Linear Parametric Model

At the rst step of the two-step approach, the global face image hg of the highresolution image h is inferred using PCA and the MAP estimator. It is assumed that the low-resolution image l obtained from h by smoothing and down-sampling loses the local feature hl : l = Ah = Ahg Therefore, Eq. (12.4) is rewritten as p(l|h) = 1 1 exp 2 Ahg l Z 2

(12.14)

= p(l|hg ).

(12.15)

Also, Eq. (12.6) is de ned by both hg and hl : h = arg max p(l|h)p(h)

= arg max p(l|hg )p(hg , hl )