Edge weightbased segmentation methods, such as normalized cut or minimum cut, require a partition number specification for their energy formulation. The number of partitions plays an important role in the segmentation overall quality. However, finding a suitable partition number is a nontrivial problem, and the numbers are ordinarily manually assigned. This is an aspect of the general partition problem, where finding the partition number is an important and difficult issue. In this paper, the edge weights instead of the pixels are partitioned to segment the images. By partitioning the edge weights into two disjoints sets, that is, cut and connect, an image can be partitioned into all possible disjointed segments. The proposed energy function is independent of the number of segments. The energy is minimized by iterating the QPBO
There are numerous approaches and applications for unsupervised image segmentation in computer vision. Many different theories are proposed for varying the roles of the unsupervised segmentation. As a low level vision problem, an image can be simplified by oversegmentation using a number of different approaches, such as modeseeking mean shift, multilevel thresholding, histogrambased neural networks, superpixel algorithms, and various graphbased methods [
Generally, the role of unsupervised segmentation falls between image simplification and full semantic segmentation, where semantically meaningful segments are expected to be found but not necessarily recognized. Segmentation is posed as an imagecoloring problem that minimizes specific energy functions. Energy functions can be optimized using stochastic methods such as deterministic annealing and stochastic clustering [
The edge weightbased segmentation methods have evolved together with graph partition problems. When edge weights are all positive, the minimum cut can be found; however, the minimum cut has bias toward smaller cuts. Adding negative edge weights can prevent the problem so the graph becomes nonsubmodular; however, the problem becomes NPhard [
For the graph theoretic segmentation and level set methods, the number of segments must be predefined. The segment number choice greatly influences the quality of segmentation, especially for a normalized cut. Nonetheless, there have been attempts to solve this problem. The number of segments can be controlled by setting the threshold value to the recursive normalized cut [
In this paper, transforming the pixel clustering problem into an edge partition problem circumvents the segment number selection problem. Edges among adjacent pixels can represent dissimilarity or similarity weights. Two edge partitions are always sufficient for pixelpartitioning problems. An edge can be in a cut set or connected set, which can then be translated into a unique segmentation, as in Figure
An image can be segmented by partitioning edges into two sets. Cut (dotted red) and connected (solid black) edge sets can be translated into a unique segmentation as in (c). However, it is also possible to have edge partitions that contradict the label assignments as in (d). By finding the image labeling that minimizes the edge partition energy, edge partitions like (d) are prevented, and a consistent image segmentation becomes possible as shown in (a) and (b).
Under the pixellabeling framework, a label number selection problem arises. Although the label number selection might seem similar to the segment number selection problem, there are subtle differences. First, pixels do not need to use all label assignments; thus, low numbers of segments are possible with large numbers of labels. Second, under the fourcolor map theorem, the maximum number of labels for twodimensional (2D) segmentation can be as low as four. The fourcolor map theorem states that any 2D map can be colored with intact borders using a maximum of four colors [
In the following sections, a new energy function is introduced for image segmentation through the edge partition. The edge partitions can uniquely define the image segmentation with the hard constraints enforced by the imagelabeling framework. Next, an energy minimization algorithm is proposed for the edge partitioning. The experimental section discusses tests of the proposed algorithm using the Berkeley image segmentation database.
Image segmentation can be viewed as a pixelpartitioning problem. Many image segmentation methods borrow their ideas from the general partitioning techniques. The
An image can be represented as a set of nodes and edges by a graph
The segmentation problem is formulated in terms of edge partitions. The edges can be partitioned into two sets
The proposed energy function breaks down into an imagelabeling problem in order to maintain the label consistency conditions of (
Given the image label state
The multilabel pairwise energy function (
Similar to the original
(
(
(
(
(
Various examples of the edge partition segmentation results using the color distance edge weights are shown in Figure
Even with the simple color distance weight, the edge partitions can produce adequate segmentation results. Some of the segmentation results from the MSRC database are shown.
The global probability of the boundary (GPB) edge detection method [
The QPBOI optimization scheme can be efficient for (a) and (b). However, for many cases, the nonsubmodular potentials are too strong, and the QPBOI optimization fails in (c) and (d). More iterations of QPBOI can improve the result; however, iterations are timeconsuming without improvement guarantees. Superpixel images are used in this study to reduce the computation time and increase the QPBOI iterations.
The proposed edge partition approach is evaluated using the popular Berkeley image database. The set contains 300 images with at least four human segment annotations per image. The three quantitative evaluation methods used are as follows: Probabilistic Rand Index (PRI) [
The evaluation methods used in this study are PRI, VoI, and BDE. PRI counts the number of consistent labels between the segmentation and the ground truth. VoI measures the segmentation randomness that cannot be explained by the ground truth. BDE is the average displacement error or the boundary pixels between two segmentation results. PRI counts the correctness in segmentation, while VoI and BDE measure the errors between the segmentation and ground truth. In the first subsection, the proposed method is evaluated against various segmentation methods. In the second subsection, the comparison between the proposed and the mergethreshold methods is demonstrated using the same edge weights.
Generally, the parameters are constant for the entire database and test methods. This evaluation includes mean shift (MShift) [
Constant parameters are maintained for the Berkeley image set and the test methods. The top ranking results are written in bold, and the rankings are in parenthesis. For segmentation methods with a
Method  PRI  VoI  BDE 

MShift [ 
0.7958  1.9725  14.41 
JSEG [ 
0.7756  2.3217  14.40 ( 
GBIS [ 
0.7139  3.3949  16.67 
NTP [ 
0.7521  2.4954  16.30 
Saliency [ 
0.7758  1.8165  16.24 
TBES [ 
0.80 ( 
1.76 ( 
— 
CtoR [ 

1.717 ( 
11.57 ( 


EPartition  0.804 ( 




NCut 
0.772  2.259  12.94 
SpecSeg 
0.8146  1.8545  12.21 
For PRI measurements, the mergethreshold method of CtoR ranks first. The proposed segmentation ranks first for VoI and BDE. The CtoR method is available to the public by the authors. The threshold value for the CtoR method was chosen to be 80 for its highest average ranking. A number of segmentation results of CtoR and of the proposed EPartition are shown in Figure
The segmentation results from Table
Image
CtoR
EPartition
CtoR and EPartition use the same edge weights; thus, their performances are similar. However, in CtoR, a mergethreshold algorithm is used for segmentation. Different thresholds among integer intervals
The segmentation evaluation (PRI, VoI, and BDE) versus threshold values is plotted for CtoR. Different threshold values give an optimal score for each segmentation evaluation approach. In contrast, the proposed EPartition is independent of the threshold values and finds the approximate optimal segmentation for all evaluation approaches.
PRI
VoI
BDE
In contrast, the edge partitioning segmentation is independent of a threshold value. Figure
In previous experiments, EPartition was shown to have competitive performance with CtoR when the optimal threshold value is handpicked for CtoR. In this section, the threshold value is trained from the Berkeley 300 set and the segmentation performances are compared to the Weizmann segmentation set [
In Table
The segmentation results for the Weizmann image set are summarized. The threshold value of CtoR is trained from the Berkeley set. Superior results are written in bold.
Method  PRI  VoI  BDE 

CtoR [ 


22.01 


EPartition  0.7169  1.225 

The segmentation results from Table
CtoR
EPartition
CtoR
EPartition
In this paper, image segmentation by edge partitioning is proposed. In contrast with previous edge weightbased segmentation methods, such as normalized cut, the proposed method is independent of the number of segments. Furthermore, compared with the previous segmentation techniques, edge partitioning remains competitive without the need for the segmentation number selection. Segmentation by edge partitioning has shown to be competitive with previous segmentation techniques in the Berkeley database. The advantage of the proposed method lies in its adaptive nature for handling edge weights without threshold values or segment number assignments.
The proposed algorithm can be extended to general partitioning problems. Four labels are sufficient when segmenting 2D images. However, for fully connected graphs, the number of labels can be arbitrarily large. If a maximum number of labels are chosen, the edge partitioning method can be incorporated into a general partition problem without designating the specific number of partitions among nodes.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by Institute for Information & Communications Technology Promotion (IITP) Grant funded by the Korea government (MSIP) (no. R0101150171, Development of Multimodality Imaging and 3D SimulationBased Integrative DiagnosisTreatment Support Software System for Cardiovascular Diseases). This work was also supported by Hankuk University of Foreign Studies Research Fund.