Registration for Variform Object of Remote-Sensing Image Using Improved Robust Weighted Kernel Principal Component Analysis

Abstract

For pre- and post-earthquake remote-sensing images, registration is a challenging task due to the possible deformations of the objects to be registered. To overcome this problem, in a previous paper, we proposed a registration method based on robust weighted kernel principal component analysis (RWKPCA) to precisely register the variform objects. Which was proved very effective in capturing the common robust kernel principal components (RKPCs) and generalized well for registration. Compared with previous paper, there are two improvements in this paper: Firstly, we developed the improved RWKPCA method from the robust loss function, and theoretically proved the robustness of the method; Secondly, a new construction of weight function by projection residual was given, which enables the great reduction of computing time. Finally, two experiments were conducted on the remote-sensing image registration in Wenchuan earthquake and change detection of Tangjiashan barrier lake, and the results showed that compared with the previous method, the registration accuracy was increased while the computational time was decreased a lot. Meanwhile, good performance on the change detection of barrier lake is obtained.

Appendices

Appendix 1: Proof of Proposition 1

Proof: Firstly, let ψ(x _i ) = w _i ϕ(x _i ), i = 1, 2, ⋯ n, ${\mathbf{C}}_{\mathbf{W}}=\frac{1}{n}{\displaystyle \sum_{i=1}^n\psi \left({\mathbf{x}}_i\right)\psi {\left({\mathbf{x}}_i\right)}^T}=\frac{1}{n}{\displaystyle \sum_{i=1}^n{w}_i^2\phi \left({\mathbf{x}}_i\right)\phi {\left({\mathbf{x}}_i\right)}^T}$, Ũ _K is the space spanned by the first k eigenvectors of C _W , then by Lemma 1 one can easily get that Ũ _K dose solve the optimization problem

$$ \begin{array}{l}{ \min}_{\mathbf{U}}\kern1.5em {\mathrm{J}}^{\perp}\left(\tilde{\mathbf{U}}\right)={{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{{\tilde{\mathbf{U}}}^{\perp }}\left(\psi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2\\ {} subject\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em \dim\ \tilde{\mathbf{U}}=k.\end{array} $$

(23)

Substituting ψ(x _i ) = w _i ϕ(x _i ) into (23), one proves that Ũ _K dose solve the following optimization problem

$$ \begin{array}{l}{ \min}_{\mathbf{U}}\kern1.5em {\mathrm{J}}^{\perp}\left(\tilde{\mathbf{U}}\right)={{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{{\tilde{\mathbf{U}}}^{\perp }}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2\\ {} subject\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em \dim\ \tilde{\mathbf{U}}=k.\end{array} $$

(24)

Having equation (24) and to prove equation (23), one only need to prove that ${{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{{\tilde{\mathbf{U}}}^{\perp }}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i{P}_{{\tilde{\mathbf{U}}}^{\perp }}\left(\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2$. Using the definition of residual and Pythagoras’s theorem, one gets

$$ \begin{array}{l}{{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{{\tilde{\mathbf{U}}}^{\perp }}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i\phi \left({\mathbf{x}}_i\right)-{P}_{\tilde{\mathbf{U}}}\Big({w}_i\phi \left({\mathbf{x}}_i\right)\right\Vert}}_2^2\hfill \\ {}={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i\phi \left({\mathbf{x}}_i\right)\right\Vert}}_2^2-{{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{\tilde{\mathbf{U}}}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2\hfill \end{array} $$

(25)

Let Ũ = [ũ ₁, ũ ₂, ⋯ ũ _k ], then one obtains

$$ {P}_{\tilde{\mathbf{U}}}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)=\left({P}_{{\tilde{\mathbf{u}}}_1}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right),{P}_{{\tilde{\mathbf{u}}}_2}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right),\cdots {P}_{{\tilde{\mathbf{u}}}_k}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right) $$

(26)

By equation (26) one can compute the projection of ϕ(x _i ) on ũ _t

$$ \begin{array}{l}{P}_{{\tilde{\mathbf{u}}}_t}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)=\left\langle {\tilde{\mathbf{u}}}_t,{w}_i\phi \left({\mathbf{x}}_i\right)\right\rangle ={w}_i\phi \left({\mathbf{x}}_i\right)\cdot {\tilde{\mathbf{u}}}_t = {w}_i\left(\phi \left({\mathbf{x}}_i\right)\cdot {\tilde{\mathbf{u}}}_t\right)\\ {}\kern4.5em ={w}_i{P}_{{\tilde{\mathbf{u}}}_t}\left(\phi \left({\mathbf{x}}_i\right)\right)\end{array} $$

(27)

Where ũ _t is the t th component of Ũ. Combining equation (26) and (27), one gets

$$ \begin{array}{l}{\left\Vert {P}_{\tilde{\mathbf{U}}}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}_2^2={\left\Vert \left({P}_{{\tilde{\mathbf{U}}}_1}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right),{P}_{{\tilde{\mathbf{u}}}_2}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right),\cdots {P}_{{\tilde{\mathbf{u}}}_k}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right)\right\Vert}_2^2\hfill \\ {}\begin{array}{lllll}\hfill & \hfill & \hfill & \hfill & ={\left\Vert \left({w}_i{P}_{{\tilde{\mathbf{U}}}_1}\left(\phi \left({\mathbf{x}}_i\right)\right),{w}_i{P}_{{\tilde{\mathbf{u}}}_2}\left(\phi \left({\mathbf{x}}_i\right)\right),\cdots {w}_i{P}_{{\tilde{\mathbf{u}}}_k}\left(\phi \left({\mathbf{x}}_i\right)\right)\right)\right\Vert}_2^2\hfill \end{array}\hfill \\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill ={\left\Vert {w}_i\left({P}_{{\tilde{\mathbf{U}}}_1}\left(\phi \left({\mathbf{x}}_i\right)\right),{P}_{{\tilde{\mathbf{u}}}_2}\left(\phi \left({\mathbf{x}}_i\right)\right),\cdots {P}_{{\tilde{\mathbf{u}}}_k}\left(\phi \left({\mathbf{x}}_i\right)\right)\right)\right\Vert}_2^2\hfill \end{array}\hfill \\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill ={\left\Vert {w}_i{P}_{\tilde{\mathbf{U}}}\left(\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}_2^2\hfill \end{array}\hfill \end{array} $$

(28)

Substituting (28) into (25) leads to

$$ \begin{array}{l}{{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{{\tilde{\mathbf{U}}}^{\perp }}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i\phi \left({\mathbf{x}}_i\right)\right\Vert}}_2^2-{{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{\tilde{\mathbf{U}}}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2\hfill \\ {}={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i\phi \left({\mathbf{x}}_i\right)\right\Vert}}_2^2-{{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i{P}_{\tilde{\mathbf{U}}}\left(\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2={\displaystyle {\sum}_{i=1}^n{w}_i^2\left({\left\Vert \phi \left({\mathbf{x}}_i\right)\right\Vert}_2^2-{\left\Vert {P}_{\tilde{\mathbf{U}}}\left(\phi \left({\mathbf{x}}_i\right)\right)\Big)\right\Vert}_2^2\right)}\hfill \\ {}={{\displaystyle {\sum}_{i=1}^n{w_i}^2\left\Vert \phi \left({\mathbf{x}}_i\right)-{P}_{\tilde{\mathbf{U}}}\left(\phi \left({\mathbf{x}}_i\right)\right)\Big)\right\Vert}}_2^2={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i\Big(\phi \left({\mathbf{x}}_i\right)-{P}_{\tilde{\mathbf{U}}}\left(\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2\hfill \\ {}={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i{P}_{{\tilde{\mathbf{U}}}^{\perp }}\left(\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2\hfill \end{array} $$

(29)

Then ${{\displaystyle {\sum}_{i=1}^n\left\Vert {P}_{{\tilde{\mathbf{U}}}^{\perp }}\left({w}_i\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2={{\displaystyle {\sum}_{i=1}^n\left\Vert {w}_i{P}_{{\tilde{\mathbf{U}}}^{\perp }}\left(\phi \left({\mathbf{x}}_i\right)\right)\right\Vert}}_2^2$ was proved, so the proposition 1 was held.

Appendix 2: Proof of Proposition 2

Proof: Firstly, we prove the symmetry of K _W . (K _W )^T = (WKW)^T = W ^T K ^T W ^T = WKW = K _W . Secondly, we prove K _W is positive semi-definite. Assume the eigenvalues of K is λ _i (1 ≤ i ≤ n), then the eigenvalues of K _W is w ² _i λ _i (1 ≤ i ≤ n), for K is positive semi-definite, all λ _i (1 ≤ i ≤ n) are greater than or equal to zero, so all w ² _i λ _i (1 ≤ i ≤ n) are also greater than or equal to zero, which proves that K _W is positive semi-definite, so K _W is also a mercer kernel.

Appendix 3

Fig. 6

Original ALOS-PALSAR intensity images and their corresponding segmented ones of Shangyou reservoir. a Pre-earthquake image acquired on February 17, 2008; b The segmented image of (a); c Post-earthquake image acquired on May 19, 2008; d The segmented image of (c)

Fig. 7

Original ALOS-PALSAR intensity images and their corresponding segmented ones of Baishuihe reservoir. a Pre-earthquake image acquired on February 17, 2008; b The segmented image of (a); c Post-earthquake image acquired on May 19, 2008; d The segmented image of (c)

Fig. 8

The ALOS AVNIR-2 images of the study area. Both images are combined by band 3 (Red color), band 2 (Green color) and band 1 (Blue color). a The pre-earthquake image acquired on Mar. 31, 2007; b The post-earthquake image acquired on May 16, 2008