Random walker algorithm

The Random Walker algorithm is an algorithm for image segmentation. In the first description of the algorithm, a user interactively labels a small number of pixels with known labels (called seeds), e.g., "object" and "background". The unlabeled pixels are each imagined to release a random walker, and the probability is computed that each pixel's random walker first arrives at a seed bearing each label, i.e., if a user places K seeds, each with a different label, then it is necessary to compute, for each pixel, the probability that a random walker leaving the pixel will first arrive at each seed. This computation may be determined analytically by solving a system of linear equations. After computing these probabilities for each pixel, the pixel is assigned to the label for which it is most likely to send a random walker. The image is modeled as a graph, in which each pixel corresponds to a node which is connected to neighboring pixels by edges, and the edges are weighted to reflect the similarity between the pixels. Therefore, the random walk occurs on the weighted graph (see Doyle and Snell for an introduction to random walks on graphs).

Although the initial algorithm was formulated as an interactive method for image segmentation, it has been extended to be a fully automatic algorithm, given a data fidelity term (e.g., an intensity prior).

The algorithm was initially published as a conference paper and later as a journal paper.

Mathematics

Although the algorithm was described in terms of random walks, the probability that each node sends a random walker to the seeds may be calculated analytically by solving a sparse, positive-definite system of linear equations with the graph Laplacian matrix, which we may represent with the variable $L$. The algorithm was shown to apply to an arbitrary number of labels (objects), but the exposition here is in terms of two labels (for simplicity of exposition).

Assume that the image is represented by a graph, with each node $v_{i}$ associated with a pixel and each edge $e_{ij}$ connecting neighboring pixels $v_{i}$ and $v_{j}$. The edge weights are used to encode node similarity, which may be derived from differences in image intensity, color, texture or any other meaningful features. For example, using image intensity $g_{i}$ at node $v_{i}$, it is common to use the edge weighting function

$w_{ij}=\exp {\left(-\beta (g_{i}-g_{j})^{2}\right)}.$

The nodes, edges and weights can then be used to construct the graph Laplacian matrix.

The random walker algorithm optimizes the energy

$Q(x)=x^{T}Lx=\sum _{e_{ij}}w_{ij}\left(x_{i}-x_{j}\right)^{2}$

where $x_{i}$ represents a real-valued variable associated with each node in the graph and the optimization is constrained by $x_{i}=1$ for $v_{i}\in F$ and $x_{i}=0$ for $v_{i}\in B$, where $F$ and $B$ represent the sets of foreground and background seeds, respectively. If we let $S$ represent the set of nodes which are seeded (i.e., $S=F\cup B$) and ${\overline {S}}$ represent the set of unseeded nodes (i.e., $S\cup {\overline {S}}=V$ where $V$ is the set of all nodes), then the optimum of the energy minimization problem is given by the solution to

$L_{{\overline {S}},{\overline {S}}}x_{\overline {S}}=-L_{{\overline {S}},S}x_{S},$

where the subscripts are used to indicate the portion of the graph Laplacian matrix $L$ indexed by the respective sets.

To incorporate likelihood (unary) terms into the algorithm, it was shown in that one may optimize the energy

$Q(x)=x^{T}Lx+\gamma \left((1-x)^{T}F(1-x)+x^{T}Bx\right)=\sum _{e_{ij}}w_{ij}\left(x_{i}-x_{j}\right)^{2}+\gamma \left(\sum _{v_{i}}f_{i}(1-x_{i})^{2}+\sum _{v_{i}}b_{i}x_{i}^{2}\right),$

for positive, diagonal matrices $F$ and $B$. Optimizing this energy leads to the system of linear equations

$\left(L_{{\overline {S}},{\overline {S}}}+\gamma F_{{\overline {S}},{\overline {S}}}+\gamma B_{{\overline {S}},{\overline {S}}}\right)x_{\overline {S}}=-L_{{\overline {S}},S}x_{S}-\gamma F_{{\overline {S}},{\overline {S}}}.$

The set of seeded nodes, $S$, may be empty in this case (i.e., ${\overline {S}}=V$), but the presence of the positive diagonal matrices allows for a unique solution to this linear system.

For example, if the likelihood/unary terms are used to incorporate a color model of the object, then $f_{i}$ would represent the confidence that the color at node $v_{i}$ would belong to object (i.e., a larger value of $f_{i}$ indicates greater confidence that $v_{i}$ belonged to the object label) and $b_{i}$ would represent the confidence that the color at node $v_{i}$ belongs to the background.

Extensions

The traditional random walker algorithm described above has been extended in several ways:

• Random walks with restart
• Alpha matting
• Threshold selection
• Soft inputs
• Run on a presegmented image
• Scale space random walk
• Fast random walker using offline precomputation

Applications

Beyond image segmentation, the Random Walker algorithm has been additionally applied to several problems in computer vision and graphics:

• Image Colorization
• Interactive rotoscoping
• Medical image segmentation
• Merging multiple segmentations
• Mesh segmentation
• Mesh denoising
• Shadow elimination
• Stereo matching

External links

• Matlab code implementing the original random walker algorithm
• Matlab code implementing the random walker algorithm with precomputation