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# 计算机代写|机器学习代写Machine Learning代考|CITS5508 Graphcuts for the Ising model

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## 计算机代写|机器学习代写Machine Learning代考|Graphcuts for the Ising model

Let us start by considering a binary MRF where the edge energies have the following form:
$$\mathcal{E}{u v}\left(x_u, x_v\right)= \begin{cases}0 & \text { if } x_u=x_v \ \lambda{u v} & \text { if } x_u \neq x_v\end{cases}$$
where $\lambda_{s t} \geq 0$ is the edge cost. This encourages neighboring nodes to have the same value (since we are trying to minimize energy). Since we are free to add any constant we like to the overall energy without affecting the MAP state estimate, let us rescale the local energy terms such that either $\mathcal{E}_u(1)=0$ or $\mathcal{E}_u(0)=0$.

Now let us construct a graph which has the same set of nodes as the MRF, plus two distinguished nodes: the source $s$ and the sink $t$. If $\mathcal{E}u(1)=0$, we add the edge $x_u \rightarrow t$ with cost $\mathcal{E}_u(0)$. Similarly, If $\mathcal{E}_u(0)=0$, we add the edge $s \rightarrow x_u$ with cost $\mathcal{E}_u(1)$. Finally, for every pair of variables that are connected in the MRF, we add edges $x_u \rightarrow x_v$ and $x_v \rightarrow x_u$, both with cost $\lambda{u, v} \geq 0$. Figure $9.4$ illustrates this construction for an MRF with 4 nodes and the following parameters:
$$\mathcal{E}1(0)=7, \mathcal{E}_2(1)=2, \mathcal{E}_3(1)=1, \mathcal{E}_4(1)=6 \lambda{1,2}=6, \lambda_{2,3}=6, \lambda_{3,4}=2, \lambda_{1,4}=1$$
Having constructed the graph, we compute a minimal $s-t$ cut. This is a partition of the nodes into two sets, $\mathcal{X}_s$ and $\mathcal{X}_t$, such that $s \in \mathcal{X}_s$ and $t \in \mathcal{X}_t$. We then find the partition which minimizes the sum of the cost of the edges between nodes on different sides of the partition:
$$\operatorname{cost}\left(\mathcal{X}s, \mathcal{X}_t\right)=\sum{x_u \in \mathcal{X}_s, x_v \in \mathcal{X}_t} \operatorname{cost}\left(x_u, x_v\right)$$
In Figure 9.4, we see that the min-cut has cost 6. Minimizing the cost in this graph is equivalent to minimizing the energy in the MRF. Hence nodes that are assigned to $s$ have an optimal state of 0 , and the nodes that are assigned to $t$ have an optimal state of 1 . In Figure 9.4, we see that the optimal MAP estimate is $(1,1,1,0)$. Thus we have converted the MAP estimation problem to a standard graph theory problem for which efficient solvers exist (see e.g., [CLR90]).

## 计算机代写|机器学习代写Machine Learning代考|Graphcuts for binary MRFs with submodular potentials

We now discuss how to extend the graphcuts construction to binary MRFs with more general kinds of potential functions. In particular, suppose each pairwise energy satisfies the following condition:
$$\mathcal{E}{u v}(1,1)+\mathcal{E}{u v}(0,0) \leq \mathcal{E}{u v}(1,0)+\mathcal{E}{u v}(0,1)$$
In other words, the sum of the diagonal energies is less than the sum of the off-diagonal energies. In this case, we say the energies are submodular (??). An example of a submodular energy is an Ising model where $\lambda_{u v}>0$. This is also known as an attractive MRF or associative MRF, since the model “wants” neighboring states to be the same.

It is possible to modify the graph construction process for this setting, and then apply graphcuts, such that the resulting estimate is the global optimum [GPS89].

## 计算机代写|机器学习代写Machine Learning代考|Graphcuts for the Ising model

$$\mathcal{E} u v\left(x_u, x_v\right)= \begin{cases}0 & \text { if } x_u=x_v \lambda u v \quad \text { if } x_u \neq x_v\end{cases}$$

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