2.1. Image Model
Let z = {zi; i = 1, …, n} be a panchromatic remote sensing image, where i is the index of pixels, n is the number of pixels, and zi is the intensity of pixel i. In statistical terms, z can be regarded as the realization of a random field Z = {Zi; i = 1, …, n}, where Zi is the random variable defined on pixel i.
In the FMM-based segmentation algorithm, the pdf of the pixel intensity
zi can be expressed as:
where
Ψi = {
αi,
Ω} is the set of model parameters for pixel
i;
αi = {
αli;
l = 1, …,
k} is the set of component weights for pixel
i;
l is the index of the component;
k is the number of components corresponding to the number of object regions;
αli represents the probability that pixel
i is assigned to the object region
l and satisfies the conditions 0 ≤
αli ≤ 1 and
;
Ω = {
Ωl;
l = 1, …,
k} is the set of component parameters; and
pli(
zi|
Ωl) is the pdf of the
zi conditional on the set of the component parameter
Ωl, which is used to mainly model the statistical distribution of the object region
l.
Equation (1) is the general formulation of mixture models, where
pli(
zi|
Ωl) can be defined by various types of distributions. For example, in GMM, its component is the Gaussian distribution with a mean
μl and variance
σl2, that is:
where
Ωl= {
μl,
σl2} is the set of parameters for component
l in GMM. Additionally, commonly used mixture models include SMM and GaMM [
22,
25], and their components are defined by the student’s t and Gamma distributions, respectively.
In this study, a new HGMM is proposed, where the components can accurately model the asymmetric, heavy-tailed, and multimodal distributions of pixel intensities in each object region. The pdf of its component can be expressed as:
where in component
l,
Ωl = {
ml,
wl,
θl} is the set of parameters;
ml is the number of elements and can be viewed as a random variable to flexibly model the complicated distributions;
wl = {
wlj;
j = 1, …,
ml} is the set of element weights;
j is the index of elements;
wlj is the weight of element
j and satisfies the conditions 0 ≤
wlj ≤ 1 and
;
θl = {
θlj;
j = 1, …,
ml} is the set of element parameters;
plji(
zi|
θlj) is the pdf of the Gaussian distribution called element;
θlj = {
μlj,
σlj2} is the set of element parameters; and
μlj and
σlj2 are the mean and variance, respectively.
Combining Equations (1) and (3), the pdf of
zi given
Ψi can be modeled by the HGMM as:
Equation (4) shows that HGMM has a distinct hierarchy. The HGMM has three layers, and each layer can be concretely expressed as follow:
1) The basic layer consists of elements, which are used to model the distribution of local sections of the object region.
2) The second layer consists of components, which are defined by the weighted sums of elements. They are used to model the statistical distributions of pixel intensities in the object regions.
3) The last layer comprises the HGMM, which is defined by the weighted sums of components, which are used to model the statistical distribution of pixel intensities in high-resolution remote sensing imagery.
Assume that the pixel intensities are statistically independent, and the joint distribution of
z given
Ψ can be written as:
where
Ψ = {
Ψi;
i = 1, …,
n} is the set of model parameters, which can be further written as
Ψ = {
α,
m,
w,
μ,
σ2};
α = {
αi;
i = 1, …,
n} is the set of component weights;
m = {
ml;
l = 1, …,
k} is the set of element numbers;
w = {
wl;
l = 1, …,
k} is the set of element weights;
μ = {
μlj;
l = 1, …,
k,
j = 1, …,
ml} is the set of means; and,
σ2 = {
σlj2;
l = 1, …,
k,
j = 1, …,
ml} is the set of variances.
To present the modeling ability of the proposed HGMM and the role of its layers, the complex histogram and its fitting curves are shown in
Figure 1. The histograms of data are shown in
Figure 1a, which presents the regions in different colors: Yellow (Region 1), green (Region 2), and blue (Region 3). The yellow histogram is asymmetric; the green histogram is asymmetric, heavy tailed, and multimodal; and the blue histogram is asymmetric and multimodal. The fitting curves of the HGMM and its components are shown in
Figure 1b, which shows that the HGMM with the new components can accurately fit the complex histograms. For example, the green histogram, which is asymmetric, heavy tailed, and multimodal, can accurately be fitted by the green curve. The fitting curves of the elements for each component are shown in
Figure 1c, which further indicates the ability of the HGMM to fit complex histograms.
2.2. Segmentation Model
In order to realize image segmentation, the posterior distribution of
Ψ given
z is viewed as the segmentation model. Following the Bayesian theorem [
29,
30], the segmentation model can be built by combining the HGMM and the prior distributions of its parameters. It can be written as:
where
p(
Ψ) is the prior distribution of
Ψ. According to the relations among model parameters,
p(
Ψ) is further written as:
The prior distributions of model parameters are defined as follows:
1)
p(
α). In order to take the spatial relations of pixels into account, MRF [
8,
9] is built on the component weights. The pdf of
αi can be defined by the Gibbs distribution [
8] given the component weights of neighboring pixels for pixel
i. Then, the prior distribution of
α can be modeled as:
where
A is the normalizing constant and is usually set to be 1;
β is a constant controlling the smoothing strength of neighboring pixels of pixel
i;
Ni is the set of indexes for neighboring pixels in the square window centered at pixel
i; and,
i’ is the index of neighboring pixels.
2)
p(
m). Assume that the number of elements
ml,
l = 1, …,
k satisfy the independent and identical truncated Poisson distribution [
15] with parameter
λ, which can be written as:
3)
p(
w|
m). Given the numbers of elements
ml,
l = 1, …,
k, the element weight vectors
wl,
l = 1, …,
k have different dimensions. Since the Dirichlet distribution [
31] is flexible in modeling the statistical distribution of multidimensional data, it can be used to model the pdf of the weight vector
wl. Assume that the element weights are independent among components. Accordingly, the prior distribution of
w given
m can be written as:
where Γ(·) is Gamma function;
δlj is the parameter of the Dirichlet distribution, which can be further written as
δ = {
δl;
l = 1, …,
k} = {
δlj;
j = 1, …,
ml}; and,
δlj.s are constants for controlling the peak position and the steepness of the distribution curves. To simplify the model solution,
δlj is set to be the same, i.e.,
δlj =
δ. Then, the symmetric Dirichlet distribution is used as the prior distribution of the element weight, which can be expressed as:
4)
p(
μ|
m) and
p(
σ|
m). Given the number of elements
ml,
μlj (
σlj) are assumed to satisfy the identical and independent Gaussian distribution with a mean
μμ (
μσ) and variance
σμ2 (
σσ2), i.e.,
μlj ~
N(
μμ,
σμ2) and
σlj ~
N(
μσ,
σσ2), where
μμ(
μσ) and
σμ2(
σσ2) are constants. The joint prior distributions of
μ and
σ can be written as:
2.3. Optimal Segmentation
To realize image segmentation, a new BDMCMC algorithm [
15,
32] is designed to simulate from the segmentation model in Equation (6), which can implement parameter sampling in various dimensions. The simulation process can be summarized as follows: Let the set of current parameters be
Ψ and the set of candidate parameters be
Ψ*, where
t is the index of iterations. The acceptance rate
a(
Ψ,
Ψ*) of
Ψ* can be calculated using the posterior distributions of
Ψ and
Ψ*. If
a(
Ψ,
Ψ*) = 1, then
Ψ(t+1) =
Ψ*; otherwise,
Ψ(t+1) =
Ψ. Four simulating operations are designed in this study, including updating the component weight, updating the element weight, updating the set of element parameters, and the birth or death an element in a component. The specific processes and the acceptance rates of each operation are described as follows:
1) Updating the component weight operation. Let the randomly selected component weight be
αli, and this is then changed to
αli +
α*, where
α*∈(−
αli, 1) is the increment weight. To satisfy the constraints of the component weights for pixel
i, their normalization is needed to change each component weight for pixel
i. As a result, the candidate component weights for pixel
i can be written as:
Then, the set of candidate component can be written as
α* = {
α1, …,
αi−1,
αi*,
αi+1, …,
αn}. Its acceptance rate can be obtained as:
2) Updating the element weight operation. Let the randomly selected element weight be
wlj, which can then be changed to
wlj +
w*, where
w*∈(−
wlj, 1) is the increment of the weight. To satisfy the constraints of the element weight in the component
l, their normalization is implemented to change each element weight for the component
l. Consequently, the candidate element weight for the component
l can be written as:
Then, the set of candidate element weights can be written as
w* = {
w1, …,
wl−1,
wl*,
wl+1, …,
wk}. Its acceptance rate can be written as:
3) Updating the parameter of the element. Let the randomly selected set of element parameters be
θlj = {
μlj,
σlj2} and the candidate set of parameters be
θlj* = {
μlj*,
σlj*2}, where
μlj* (
σlj*) is randomly generated from the Gaussian distribution with the mean
μlj (
σlj), and the standard deviation
εμ (
εσ) is set to 0.5. The set of candidate parameters in component
l can be written as
θl* = {
θl1, …,
θlj−1,
θlj*,
θlj+1, …,
}. Then, the set of candidate element parameters can be written as
θ* = {
θ1, …,
θl−1,
θl*,
θl+1, …,
θk}. Its acceptance rate can be calculated as:
4) Birth or death of an element in a component. The index of component
l is randomly selected from {1, …,
k}, and its number of elements is
ml. When adding an element in the component
l, the candidate number of elements is
ml* =
ml + 1, and correspondingly add
and
, where the added mean and variance are generated by their prior distribution. The element weights satisfy the condition
. Then, the candidate set of element weights can be written as:
.
The set of candidate element parameters can be written as
θl*
= {
θl1, …,
,
}. The acceptance rate of adding an element can be written as
a(
m,
m*) = min(1,
R), where
R is written as:
When
ml > 2, the deletion of an element can be carried out. In this case, the index of the deleted element
j is randomly selected from {1, …,
ml}, and correspondingly delete
wlj and
θlj. The new element weights of component
l can then be written as {
wl1, …,
wlj−1,
wlj+1, ….,
}. To satisfy the constraints of element weights, the candidate set of element weights can be written as:
The candidate set of element parameters can be written as θl* = {θl1*, …, θlj−1*, θlj*, …, } = {θl1, …, θlj−1, θlj+1, …, }. The candidate number of elements is ml* = ml −1. The acceptance rate of element deletion can be written as a(m, m*) = min(1, 1/R), given that this operation is antithetical with element addition.
In iterations, the above operations are carried out in sequence. To realize the image segmentation, the pixel label can be obtained by maximizing the component weights, which can be written as:
where
ci is the label of pixel
i, and
c = {
ci;
i = 1, …,
n} is the segmented result of the given image.
The proposed HGMM based segmentation algorithm is summarized in
Table 1.