Abstract
Learning partial differential equations (LPDEs) from training data for particular tasks has been successfully applied to many image processing problems. In this paper, we propose a more effective LPDEs model for vector-valued image tasks. The PDEs are also formulated as a linear combination of fundamental differential invariants, but have several distinctions. First, we simplify the current LPDEs system by omitting a PDE which works as an indicate function in current ones. Second, instead of using \(L_2\)-norm, we use the \(L_1\)-norm to regularize the coefficients with respect to the fundamental differential invariants. Third, as the objective function is not smooth, we resort to the alternating direction method to optimize it. We illustrate the properties of our LPDEs system by several examples in denoising and demosaicking of RGB color images. The experiments demonstrate the advantage of the proposed method over other PDE-based methods.
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Notes
The images are padded with zeros of several pixels width around them, so that the Dirichlet boundary conditions \(u_{m}(x,y,t)=0, v_{m}(x,y,t)=0, (x,y,t)\in \varGamma\), are naturally fulfilled.
The images are padded with zeros of several pixels width around them such that the Dirichlet boundary conditions \(u^c_{m}(x,y,t)=0, (x,y,t)\in \varGamma\), are naturally fulfilled.
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Acknowledgements
The authors thank the NSFC support (Nos. 61471369 and 61503396) and the open Research Foundation of State Key Laboratory of Astronautic Dynamics (Nos. 2013ADL-DW0101 and 2014ADL-DW0102).
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Appendix
Appendix
When each of them gets the minimum, the functional \(J(\mathbf {u}_m,\mathbf {a})\) (8) get the minimum. We simply write \(u^c_m\) as \(\varphi\) and fix \(u^k_m\) \((k\ne c)\) when we derive the formulation next. We divide \(J(\varphi ,\mathbf {a})\) into two parts
For the first part, we have
For the second part, we first define \(F(\varphi ,\mathbf {a})=\sum _{i=0}^{36}a^c_i (t)\mathbf {inv}_i (\varphi )\), Then, we get
where \(Q=\varOmega \times [0,T]\) and \(\text {d}Q={\text { d}}\varOmega {\text { d}}t\).
We first compute \(F(\varphi +\varepsilon \delta \varphi ,\mathbf {a})-F(\varphi ,\mathbf {a})\), and it equals
where
From the above formulations, we can get
As the perturbation \(\delta \varphi\) should satisfy that \(\delta \varphi |_{\varGamma }=0\) and \(\delta \varphi |_{t=0}=0\). Integrating by parts, we have
If \(\frac{\mathrm {D}L}{\mathrm {D}\varphi }\) exists, the boundary conditions must satisfy the equation as follows
Then, we get the G\(\hat{a}\)teaux derivative
When the functional gets the minimum, it should satisfy the follow PDE with the initial condition \(u^c_m|_{t=0}=I^c_m\),
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Jiao, Y., Pan, X., Zhao, Z. et al. Learning sparse partial differential equations for vector-valued images. Neural Comput & Applic 29, 1205–1216 (2018). https://doi.org/10.1007/s00521-016-2623-y
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DOI: https://doi.org/10.1007/s00521-016-2623-y