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SURE-Type Functionals as Criteria for Parametric PSF Estimation

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Abstract

Point spread function (PSF) estimation plays an important role in blind image deconvolution. This paper proposes two novel criteria for parametric PSF estimation, based on Stein’s unbiased risk estimate (SURE), namely, prediction-SURE and its variant. We theoretically prove the SURE-type functionals incorporating exact (complementary) smoother filtering as the valid criteria for PSF estimation. We also provide the theoretical error analysis for the regularizer approximations, by which we show that the proposed frequency-adaptive regularization term yields more accurate PSF estimate than others. In particular, the proposed SURE-variant enables us to avoid estimation of noise variance, which is a key advantage over the traditional SURE-like functional. Finally, we propose an efficient algorithm for the minimizations of the criteria. Not limited to the examples we show in this paper, the proposed SURE-based framework has a great potential for other imaging applications, provided the parametric PSF form is available.

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Notes

  1. The last term \(\Vert \mathbf {x}\Vert ^2\) of (3) is a constant irrelevant to the optimization of function \(\mathbf {f}\).

  2. Oracle means that this criterion is not accessible in practice, due to the unknown \(\mathbf {x}\) in (4).

  3. Oracle means that this criterion is not accessible in practice, due to the unknown \(\mathbf {H}_0\mathbf {x}\) in (7).

  4. Following the convention of [14], we refer to estimation of \(\mathbf {x}\) as estimation, and to estimation of \(\mu =\mathbf {H}_0\mathbf {x}\) as prediction.

  5. The optimal solution \(\mathbf {s}^\star \) may not be unique. The uniqueness of the solution depends on the parametric form of PSF.

  6. The practical computation can be fully performed in Fourier domain. The introduction of matrix \(\mathbf {Q}\) is for sake of concise expression by linear algebra language.

  7. The last term \(\sigma ^2\), though unknown, is a constant irrelevant to the minimization procedure.

  8. The terminology jinc is due to the structural similarity to sinc function [6].

  9. BSNR (blurr signal-to-noise ratio) is defined as \({\text {BSNR}}=10\log _{10} \Big (\frac{\Vert \mathbf {H}_0\mathbf {x}-{\text {mean}}(\mathbf {H}_0\mathbf {x})\Vert ^2}{N\sigma ^2}\Big )\) [24, 34].

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61401013). The authors would like to thank the coordinating editor and anonymous reviewers for their insightful comments, and also obliged to Hanjie PAN for many useful discussions on this paper.

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Authors and Affiliations

  1. National Key Laboratory of Science and Technology on Test Physics and Numerical Mathematics, Beijing, 100076, China

    Feng Xue, Jiaqi Liu, Shenghai Jiao, Shengdong Liu, Min Zhao & Zhenhong Niu

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  1. Feng Xue
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  3. Shenghai Jiao
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Correspondence to Feng Xue.

Appendices

Appendix 1: Proof of Theorem 3

Proof

Based on Lemma 1, the prediction-SURE can be written as

$$\begin{aligned} \epsilon =\frac{1}{N} \big \Vert \mathbf {U}\mathbf {y}-\mathbf {y}\big \Vert ^2 +\frac{2\sigma ^2}{N} {\mathrm {div}}_{\mathbf {y}}\big (\mathbf {U}\mathbf {y}\big ) -\sigma ^2. \end{aligned}$$

We omit the subscript \(\mathbf {U}_{\mathbf {H},\lambda }\) as \(\mathbf {U}\) for brevity in this proof.

Now, we are going to compute the divergence term—\({\mathrm {div}}_{\mathbf {y}}\big (\mathbf {U}\mathbf {y}\big )\). Notice that the matrix \(\mathbf {U}\) is the function of \(\mathbf {y}\), hence, \({\mathrm {div}}_{\mathbf {y}}\big (\mathbf {U}\mathbf {y}\big )\ne {\mathrm {Tr}}(\mathbf {U})\).

First, by definition of divergence term, we have

$$\begin{aligned} {\mathrm {div}}_{\mathbf {y}}\big (\mathbf {U}\mathbf {y}\big ) ={\mathrm {Tr}}(\mathbf {U})+\bigg ( \underbrace{\frac{\partial \mathrm {diag}(\mathbf {U})}{\partial \mathbf {y}}}_{\alpha }\bigg )^\mathrm {T}\mathbf {y}, \end{aligned}$$
(28)

where the vector \(\mathrm {diag}(\mathbf {U})\in \mathbf {R}^N\) consists of the diagonal element \(U_{n,n}\) of matrix \(\mathbf {U}\):

$$\begin{aligned} \frac{\partial \mathrm {diag}(\mathbf {U})}{\partial \mathbf {y}} =\left[ \frac{\mathrm {d}U_{1,1}(\mathbf {y})}{\mathrm {d}y(0)}, \frac{\mathrm {d}U_{2,2}(\mathbf {y})}{\mathrm {d}y(1)}, \ldots , \frac{\mathrm {d}U_{N,N}(\mathbf {y})}{\mathrm {d}y(N-1)}\right] ^\mathrm {T}\end{aligned}$$

Notice that under periodic boundary condition, \(\mathbf {U}\) is circulant matrix, whose diagonal element is a constant given by

$$\begin{aligned}&U_{n,n}(\mathbf {y})=\underbrace{u_N(0)}_\text {filter}\\&\quad =\frac{1}{N}\sum _{k=0}^{N-1} \underbrace{\frac{|H(k)|^2}{|H(k)|^2 +\frac{\lambda }{|Y(k)|^2}}}_{U(k)},\qquad \mathrm {for}\ n=0,1,...,N-1. \end{aligned}$$

The second equality is from inverse Fourier transform. Now, we consider the n-th element of \(\alpha \):

$$\begin{aligned}&\alpha _n=\frac{\partial U_{n,n}(\mathbf {y})}{\partial y(n)} =\frac{\partial U_{n,n}(\mathbf {y})}{\partial |Y(k)|^2} \cdot \frac{\partial |Y(k)|^2}{\partial y(n)} \\&\quad =\frac{1}{N}\sum _{k=0}^{N-1} \frac{\partial \frac{|H(k)|^2}{|H(k)|^2 +\frac{\lambda }{|Y(k)|^2}}}{\partial |Y(k)|^2} \cdot \frac{\partial |Y(k)|^2}{\partial y(n)}, \end{aligned}$$

where

$$\begin{aligned} \frac{\partial \displaystyle \bigg (\frac{|H(k)|^2}{|H(k)|^2 +\frac{\lambda }{|Y(k)|^2}}\bigg )}{\partial |Y(k)|^2} =\frac{|H(k)|^2\lambda }{\Big (|H(k)|^2 +\frac{\lambda }{|Y(k)|^2}\Big )^2\cdot |Y(k)|^4} \end{aligned}$$

and for any fixed n:

$$\begin{aligned}&\frac{\partial |Y(k)|^2}{\partial y(n)}= \frac{\partial \displaystyle \bigg |\sum _{l=0}^{N-1}y(l) e^{-j\frac{2\pi kl}{N}}\bigg |^2}{\partial y(n)}\\&\quad = \frac{\partial \displaystyle \Bigg ( \bigg |\sum _{l=0}^{N-1}y(l)\cos \frac{2\pi kl}{N}\bigg |^2 + \bigg |\sum _{l=0}^{N-1}y(l)\sin \frac{2\pi kl}{N}\bigg |^2 \Bigg )}{\partial y(n)}, \end{aligned}$$

where the two terms are

$$\begin{aligned}&\frac{\partial \displaystyle \bigg (\sum _{l=0}^{N-1}y(l)\cos \frac{2\pi kl}{N}\bigg )^2}{\partial y(n)}\\&\quad =2\cos \frac{2\pi nk}{N}\bigg ( \sum _{l=0}^{N-1}y(l)\cos \frac{2\pi lk}{N}\bigg ) \\&\quad =2\cos \frac{2\pi nk}{N}{\mathcal {R}}\{Y(k)\} \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial \displaystyle \bigg (\sum _{l=0}^{N-1}y(l)\sin \frac{2\pi kl}{N}\bigg )^2}{\partial y(n)}\\&=2\sin \frac{2\pi nk}{N}\bigg ( \sum _{l=0}^{N-1}y(l)\sin \frac{2\pi lk}{N}\bigg ) \\&=-2\sin \frac{2\pi nk}{N}{\mathcal {I}}\{Y(k)\} \end{aligned}$$

Hence, \(\alpha _n\) becomes

$$\begin{aligned} \alpha _n= & {} \frac{2}{N}\sum _{k=0}^{N-1} \underbrace{\frac{|H(k)|^2\lambda }{\Big (|H(k)|^2 +\frac{\lambda }{|Y(k)|^2}\Big )^2\cdot |Y(k)|^4}} _{V(k): \text {real-valued}}\\&\quad \cdot \bigg ({\mathcal {R}}\big \{Y(k)\big \}\cos \frac{2\pi nk}{N} \\&-{\mathcal {I}}\big \{Y(k)\big \}\sin \frac{2\pi nk}{N}\bigg )\\= & {} \frac{2}{N}\sum _{k=0}^{N-1} {\mathcal {R}}\big \{\underbrace{V(k)Y(k)}_{G(k)}\big \} \cos \frac{2\pi nk}{N} \\&-\frac{2}{N}\sum _{k=0}^{N-1} {\mathcal {I}}\big \{\underbrace{V(k)Y(k)}_{G(k)}\big \} \sin \frac{2\pi nk}{N}\\= & {} 2{\mathcal {R}}\{g(n)\}, \end{aligned}$$

where g(n) is the inverse Fourier transform of \(G(k)=V(k)Y(k)\): \(g(n)=\frac{1}{N}\sum _{k=0}^{N-1} G(k)e^{j\frac{2\pi nk}{N}}\). Since V(k) is real-valued, and Y(k) is Hermitian symmetric, g(n) is real-valued. Thus, we have \(\alpha _n=2{\mathcal {R}}\{g(n)\}=2g(n)\).

Finally, the second term of (28) becomes

$$\begin{aligned} \alpha ^\mathrm {T}\mathbf {y}=2(\mathbf {F}G)^\mathrm {T}\mathbf {F}Y =2G^\mathrm {T}\mathbf {F}^\mathrm {T}\mathbf {F}Y=2G^\mathrm {T}Y\\ =\sum _{k=0}^{N-1}V(k)Y^*(k)Y(k) =\sum _{k=0}^{N-1}V(k)|Y(k)|^2\\ =\sum _{k=0}^{N-1}\underbrace{ \frac{|H(k)|^2\lambda }{\Big (|H(k)|^2 +\frac{\lambda }{|Y(k)|^2}\Big )^2\cdot |Y(k)|^2}}_{Q(k)}, \end{aligned}$$

where \(\mathbf {F}\) is 1-D DFT matrix. The proof is completed. \(\square \)

Appendix 2: Proof of Corollary 1

Proof

The proof is very similar to that of Theorem 3 (see Appendix 1). The only difference from one-dimensional case is how to compute \(\frac{\partial |Y(k_1,k_2)|^2}{\partial y(n_1,n_2)}\):

$$\begin{aligned} \frac{\partial |Y(k_1,k_2)|^2}{\partial y(n_1,n_2)}= \frac{\partial \displaystyle \Bigg |\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)e^{-j\frac{2\pi k_1l_1}{M}} e^{-j\frac{2\pi k_2l_2}{N}}\Bigg |^2}{\partial y(n_1,n_2)}. \end{aligned}$$

The numerator is

$$\begin{aligned}&\Bigg |\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)e^{-j\frac{2\pi k_1l_1}{M}} e^{-j\frac{2\pi k_2l_2}{N}}\Bigg |^2 \nonumber \\&=\Bigg [\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)\Big (\underbrace{\cos \frac{2\pi k_1l_1}{M} \cos \frac{2\pi k_2l_2}{N} -\sin \frac{2\pi k_1l_1}{M} \sin \frac{2\pi k_2l_2}{N}}_{A(l_1,l_2)}\Big )\Bigg ]^2 \nonumber \\&\quad +\Bigg [\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)\Big (\underbrace{\cos \frac{2\pi k_1l_1}{M} \sin \frac{2\pi k_2l_2}{N} +\sin \frac{2\pi k_1l_1}{M} \cos \frac{2\pi k_2l_2}{N}}_{B(l_1,l_2)}\Big )\Bigg ]^2.\nonumber \\ \end{aligned}$$
(29)

The derivatives of both the terms of (29) are

$$\begin{aligned}&\frac{\partial \displaystyle \bigg [\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)A(l_1,l_2)\bigg ]^2}{\partial y(n_1,n_2)} \\&\quad =2\bigg ( \underbrace{\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)A(l_1,l_2)}_{{\mathcal {R}}\{Y(k_1,k_2)\}}\bigg )\\&\qquad \times \Big ( \underbrace{\cos \frac{2\pi k_1n_1}{M} \cos \frac{2\pi k_2n_2}{N} -\sin \frac{2\pi k_1n_1}{M} \sin \frac{2\pi k_2n_2}{N}}_{A(n_1,n_2)}\Big ) \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial \displaystyle \bigg [\sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)B(l_1,l_2)\bigg ]^2}{\partial y(n_1,n_2)} \\&\quad =2\bigg ( \underbrace{ \sum _{l_1=0}^{M-1}\sum _{l_2=0}^{N-1} y(l_1,l_2)B(l_1,l_2)}_{-{\mathcal {I}}\{Y(k_1,k_2)\}}\bigg )\\&\qquad \times \Big ( \underbrace{\cos \frac{2\pi k_1n_1}{M} \sin \frac{2\pi k_2n_2}{N} +\sin \frac{2\pi k_1n_1}{M} \cos \frac{2\pi k_2n_2}{N}}_{B(n_1,n_2)}\Big ). \end{aligned}$$

Hence, \(\alpha _{n_1,n_2}\) is

$$\begin{aligned} \alpha _{n_1,n_2}= & {} \frac{2}{MN}\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} \underbrace{ \frac{|H(k_1,k_2)|^2\lambda }{\Big (|H(k_1,k_2)|^2 +\frac{\lambda }{|Y(k_1,k_2)|^2}\Big )^2 \cdot |Y(k_1,k_2)|^4} }_{V(k_1,k_2): \text {real-valued}}\\&\cdot \Big ({\mathcal {R}}\{Y(k_1,k_2)\}A(n_1,n_2) -{\mathcal {I}}\{Y(k_1,k_2)\}B(n_1,n_2)\Big )\\= & {} \frac{2}{MN}\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} {\mathcal {R}}\big \{\underbrace{V(k_1,k_2)Y(k_1,k_2)} _{G(k_1,k_2)}\big \}A(n_1,n_2)\\&-\frac{2}{MN}\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} {\mathcal {I}}\big \{\underbrace{V(k_1,k_2)Y(k_1,k_2)} _{G(k_1,k_2)}\big \}B(n_1,n_2)\\= & {} 2{\mathcal {R}}\big \{g(n_1,n_2)\big \}, \end{aligned}$$

where \(g(n_1,n_2)\) is the inverse Fourier transform of \(G(k_1,k_2)=V(k_1,k_2)Y(k_1,k_2)\): \(g(n_1,n_2)=\frac{1}{MN}\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} G(k_1,k_2)e^{j\frac{2\pi n_1k_1}{M}} e^{j\frac{2\pi n_2k_2}{N}}\). Since \(V(k_1,k_2)\) is real-valued, and \(Y(k_1,k_2)\) is Hermitian symmetric, \(g(n_1,n_2)\) is real-valued. Thus, we have \(\alpha _{n_1,n_2}=2{\mathcal {R}}\{g(n_1,n_2)\} =2g(n_1,n_2)\).

Finally, the second term of (28) becomes

$$\begin{aligned} \alpha ^\mathrm {T}\mathbf {y}&=2(\mathbf {F}G)^\mathrm {T}\mathbf {F}Y =2G^\mathrm {T}\mathbf {F}^\mathrm {T}\mathbf {F}Y=2G^\mathrm {T}Y\\&=\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} V(k_1,k_2)Y^*(k_1,k_2)Y(k_1,k_2)\\&=\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} V(k_1,k_2)|Y(k_1,k_2)|^2\\&=\sum _{k_1=0}^{M-1}\sum _{k_2=0}^{N-1} \underbrace{ \frac{|H(k_1,k_2)|^2\lambda }{\Big (|H(k_1,k_2)|^2 +\frac{\lambda }{|Y(k_1,k_2)|^2}\Big )^2 \cdot |Y(k_1,k_2)|^2}}_{Q(k_1,k_2)}, \end{aligned}$$

where \(\mathbf {F}\) is 2-D DFT matrix. The proof is completed. \(\square \)

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Xue, F., Liu, J., Jiao, S. et al. SURE-Type Functionals as Criteria for Parametric PSF Estimation. J Math Imaging Vis 54, 78–105 (2016). https://doi.org/10.1007/s10851-015-0590-z

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