Application of Image Processing in Ice–Structure Interaction

Introduction

The understanding of Arctic physical processes and sustainable exploration, exploitation, and management of Arctic resources require more detailed, precise, and continuous measurements of sea ice parameters. Because various types of ice are in the ice-covered regions and the sizes of the ice floes can range from about 1 meter to kilometers, the temporally and spatially continuous field observations of sea ice are necessary for marine activities. One of the best ways of observing the ice conditions in the oceans is by using aerial or nautical imagery. The use of cameras as sensors on mobile sensor platforms (e.g., unmanned vehicles) will aid the development of sea ice observation. It has the potential of continuous measurements with high precision, which is particularly important for providing detailed localized information of sea ice to ensure safe operations of structures in ice-covered regions (Haugen et al. 2011).

The collected ice images or videos must be analyzed by...

Appendices

Appendix A Traditional and GVF Snake Algorithms

Traditional Snake Algorithm

A traditional snake is a curve C(s) = (x(s),y(s)) with the normalized arc length s ∈ [0,1] that moves through the spatial domain of an image to minimize the sum of the internal and external energy, given by

$$ \mathbf{E}={\int}_0^1\left({\mathbf{E}}_{\mathrm{int}}\left(\mathbf{C}(s)\right)+{\mathbf{E}}_{\mathrm{ext}}\left(\mathbf{C}(s)\right)\right)\mathrm{d}s $$

(11)

where E_int is the internal energy

$$ {\mathbf{E}}_{\mathrm{int}}=\frac{1}{2}\left(\alpha {\left|{\mathbf{C}}^{\prime }(s)\right|}^2+\beta {\left|{\mathbf{C}}^{{\prime\prime} }(s)\right|}^2\right) $$

(12)

where α and β are weight parameters that control the snake’s tension and rigidity, respectively. C′(s) denotes the first derivatives of C(s) with respect to s, making the snake act as a membrane, and C″(s) denotes the second derivatives, making the snake act as a thin plate.

E _ext is the external energy defined in the image domain. It attracts snakes to salient features in the image, such as boundaries. To find boundaries in a grayscale image, I(x, y), the image gradient is typically chosen as the external energy (Kass et al. 1988):

$$ {\mathbf{E}}_{\mathrm{ext}}=-{\left|\nabla \mathbf{I}\left(x,y\right)\right|}^2 $$

(13)

where $\nabla \mathbf{I}\left(x,y\right)=\left(\frac{\partial \mathbf{I}}{\partial x},\frac{\partial \mathbf{I}}{\partial y}\right)$ is the image gradient that represents a directional change in the brightness of the image with the gradient angle $\theta =\arctan \left(\frac{\partial \mathbf{I}}{\partial y}/\frac{\partial \mathbf{I}}{\partial x}\right)$. When also considering the image noise, the external energy is defined as (Kass et al. 1988)

$$ {\mathbf{E}}_{\mathrm{ext}}=-{\left|\nabla {\mathbf{G}}_{\sigma}\left(x,y\right)\ast \mathbf{I}\Big(x,y\Big)\right|}^2 $$

(14)

where ∇G_σ(x, y) is a two-dimensional Gaussian function with a standard deviation σ and “∗” denotes convolution.

To minimize the energy E, a snake must satisfy the Euler equation:

$$ \alpha {\mathbf{C}}^{{\prime\prime} }(s)-\beta {{\mathbf{C}}^{{\prime\prime}}}^{{\prime\prime} }(s)-\nabla {\mathbf{E}}_{\mathrm{ext}}=0. $$

(15)

Let F_int = αC^″(s) − βC^″″(s) denote the internal force and F_ext = − ∇ E_ext denote the external force. Then Eq. 15 can be written as the force balance:

$$ {\mathbf{F}}_{\mathrm{int}}+{\mathbf{F}}_{\mathrm{ext}}=0. $$

(16)

The internal and external forces are defined such that the snake will conform to an object boundary (or other desired features) within an image. The internal force F_int discourages stretching and bending, while the external potential force F_ext pulls the snake toward the desired image boundaries. Eq. 16 implies that the initial curve given in the snake algorithm will move under the influence of internal forces from the curve itself and external forces computed from the image data until the internal and external forces reach equilibrium.

To find a solution for Eq. 15, C(s) is treated as a discrete system of normalized arc length s and time t:

$$ \frac{\partial \mathbf{C}\left(s,t\right)}{\partial t}=\alpha {\mathbf{C}}^{{\prime\prime}}\left(s,t\right)-\beta {{\mathbf{C}}^{{\prime\prime}}}^{{\prime\prime}}\left(s,t\right)-\nabla {\mathbf{E}}_{\mathrm{ext}}. $$

(17)

When the solution C(s,t) becomes stationary, $\frac{\partial \mathbf{C}\left(s,t\right)}{\partial t}$ tends to zero, the energy E reaches a minimum, and the curve converges toward the target boundary.

The traditional snake algorithm is able to detect weak boundaries. However, there are two key limitations: the capture range of the external force fields is limited, and it is difficult for the snake to progress into boundary concavities. The traditional snake algorithm is, therefore, sensitive to the initial contour, and the initial contour should be somewhat close to the true boundary.

GVF Snake Algorithm

To overcome the limitations of the traditional snake algorithm, the GVF snake algorithm introduces a spatial diffusion of the gradient of an edge map (which is derived from the image data) to expand the capture range of external force fields from boundary regions to homogeneous regions (Xu and Prince 1998).

The GVF is defined to be the vector field v(x, y) = (u(x, y), v(x, y)) that minimizes the energy functional:

$$ \upepsilon =\int \int \left[\mu \left({u}_x^2+{u}_y^2+{v}_x^2+{v}_y^2\right)+{\left|\nabla f\right|}^2{\left|\mathbf{v}-\nabla f\right|}^2\right]\mathrm{d}x\mathrm{d}y $$

(18)

where u_x, u_y, v_x, and v_y are the derivatives of the vector field, μ is a parameter that controls the balance between the first and second term in the integrand, and f is an edge map (which could be the image gradient |∇I(x, y)|²) that is larger near the edges of objects in the image.

In Eq. 18, ∣ ∇ f∣ becomes large close to the object boundaries, in which case the second term dominates the integrand and is minimized by v = ∇ f. Otherwise, ∣ ∇ f∣ is small, and the first term dominates the integrand to ensure that the external force field varies slowly and still acts in the homogeneous regions.

The GVF field can be found by solving the Euler equations:

$$ \mu {\nabla}^2u-\left(u-{f}_x\right)\left({f}_x^2+{f}_y^2\right)=0 $$

(19a)

$$ \mu {\nabla}^2v-\left(v-{f}_y\right)\left({f}_x^2+{f}_y^2\right)=0. $$

(19b)

A solution to Eq. 19a and Eq. 19b can be obtained by introducing a time variable, t, and finding the steady-state solution of the following partial differential equations:

$$ {\displaystyle \begin{array}{l}{u}_t\left(x,y,t\right)=\mu {\nabla}^2u\left(x,y,t\right)-\Big(u\left(x,y,t\right)-\\ {}\quad \quad \quad {f}_x\left(x,y\right)\Big)\left({f}_x{\left(x,y\right)}^2+{f}_y{\left(x,y\right)}^2\right)\end{array}}. $$

(20a)

$$ {\displaystyle \begin{array}{l}{v}_t\left(x,y,t\right)=\mu {\nabla}^2v\left(x,y,t\right)-\Big(v\left(x,y,t\right)-\\ {}\quad \quad \quad {f}_y\left(x,y\right)\Big)\left({f}_x{\left(x,y\right)}^2+{f}_y{\left(x,y\right)}^2\right)\end{array}}. $$

(20b)

Compared to the external force field in the traditional snake model having only f_x and f_y, the new vector fields, u and v in the GVF, are derived using an iterative method to find a solution for f_x and f_y. The result is that the capture range is effectively enlarged, as seen Fig. 21. Therefore, the GVF snake releases the requirements of the initial contour, so that the initial contour no longer needs to be as close to the true boundary.

Fig. 21

External forces

Appendix B Distance Transform

Given a binary image f (x, y), whose elements only have values of “0” and “1,” the pixels with a value of “0” indicate the background, while the pixels with a value of “1” indicate the object. Let B = {(x, y)|f (x,y) = 0} be the set of background pixels and O = {(x,y)|f (x, y) = 1} be the set of object pixels. The distance transform of a binary image f, D(x, y) is the minimum distance from each pixel in f to the background B, that is:

$$ \mathbf{D}\left(x,y\right)=\left\{\begin{array}{ll}0& \mathrm{if}\;\left(x,y\right)\in B\\ {}{\min}_{b\in B}d\left[\left(x,y\right),b\right]& \mathrm{if}\left(x,y\right)\in O\end{array}\right. $$

(21)

where d[(x, y), b] is some distance measure between pixel (x, y) and b (Rosenfeld and Pfaltz 1968).

Appendix C Dilation and Erosion

A dilation operation “grows” or “thickens” objects by adding pixels to the object boundaries. The mathematical definition of dilation of A by B is defined as follows (Gonzalez et al. 2003):

$$ A\oplus B=\left\{z|{\left(\hat{B}\right)}_z\cap A\ne \varnothing \right\} $$

(22)

where (B)_z is the translation of B by the point z = (z₁, z₂), defined as

$$ {(B)}_z=\left\{b+z|b\in B\right\} $$

(23)

and $\hat{B}$ is the reflection of B, defined as

$$ \hat{B}=\left\{w|-w\in B\right\}. $$

(24)

The output of dilation is a binary image having a value of 1 at each location of the origin of the structuring element, such that the reflected and translated structuring element overlaps at least one 1-valued pixel in the input image.

An erosion operation “shrinks” or “thins” objects by removing pixels on object boundaries. The mathematical definition of erosion A by B is as follows (Gonzalez et al. 2003):

$$ A\ominus B=\left\{z|{(B)}_z\cap {A}^c\ne \varnothing \right\} $$

(25)

where A^c is the complement of A (0-valued pixels set to 1-valued and 1-valued pixels set to 0-valued, for a binary image), defined as

$$ {A}^c=\left\{w|w\notin A\right\}. $$

(26)

The output of erosion is a binary image having a value of 1 at each location of the origin of the structuring element, such that the element overlaps only 1-valued pixels of the input image (i.e., it does not overlap any of the image background).