An Optimal Graph Theoretic Approach to Data Clustering: Theory and Its Application to Image Segmentation

Zhenyu WuR. Leahy

article1993TPAMI1,331 citations

Develops a scalable graph-theoretic clustering method using subgraph condensation and equivalent trees to find globally optimal minimum cuts, guaranteeing closed boundary contours in image segmentation across hundreds of thousands of vertices.

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Data clustering and image segmentation are critical tools in computational data analysis, computer vision, and medical imaging. Standard clustering algorithms often struggle to find globally optimal partitions without prohibitive computational cost, while conventional edge detection methods either fail to form closed, contiguous boundaries or misplace region contours. The article addresses this challenge by presenting a globally optimal graph-theoretic data clustering framework and demonstrating its practical application to image segmentation.

The primary objective of the article is to establish a network flow-based clustering methodology that minimizes inter-subgraph similarity and to demonstrate an efficient hierarchical implementation capable of scaling to very large datasets, such as full-resolution images. The framework represents data points or image pixels as vertices in an undirected graph connected by arcs weighted by similarity. By leveraging network flow theory and the Gomory-Hu algorithm, partitioning the graph into distinct clusters corresponds to identifying minimum cuts. To overcome the computational bottleneck of running network flow algorithms on graphs with tens or hundreds of thousands of vertices, the authors introduce a hierarchical subgraph condensation technique that prunes unneeded high-capacity cuts using local processing, preserving global optimality while reducing processing time from over 12 hours to roughly 10 minutes.

The article demonstrates several key findings. First, graph partitioning via minimum cuts achieves a globally optimal cluster configuration that minimizes the maximum flow between subgraphs and produces a natural, nested sequence of optimal partitions for varying numbers of clusters. Second, the hierarchical condensation algorithm allows the framework to scale to massive graphs containing several hundred thousand vertices without losing mathematical optimality. Third, when applied to image segmentation—such as magnetic resonance brain scans and aerial photographs—the method reliably finds closed, thin edge contours along true object boundaries while naturally suppressing weak or isolated edges, outperforming traditional zero-crossing operators.

These findings indicate that complex segmentation and clustering tasks can achieve global mathematical optimality within realistic computing constraints. In practical applications like medical imaging, this approach enables more reliable isolation of critical anatomical structures and lesions without manual edge linking or thinning. Next steps supported by the article include integrating domain-specific prior information into the capacity functions and using the segmented regions as inputs to automated tissue labeling algorithms. The main operational constraints involve selecting appropriate edge-mask parameters, capacity functions, and minimum cluster size thresholds to avoid generating small, unclassified boundary fragments.

University of Pennsylvania University of Southern California bipartite graphs data clustering optimal solutions data clustering image segmentation graph-theoretic approach

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Abstract

A novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated. The data to be clustered are represented by an undirected adjacency graph G with arc capacities assigned to reflect the similarity between the linked vertices. Clustering is achieved by removing arcs of G to form mutually exclusive subgraphs such that the largest inter-subgraph maximum flow is minimized. For graphs of moderate size (~ 2000 vertices), the optimal solution is obtained through partitioning a flow and cut equivalent tree of G, which can be efficiently constructed using the Gomory-Hu algorithm. However for larger graphs this approach is impractical. New theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size. This algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices. The new clustering algorithm is applied to the image segmentation problem. The segmentation is achieved by effectively searching for closed contours of edge elements (equivalent to minimum cuts in G), which consist mostly of strong edges, while rejecting contours containing isolated strong edges. This method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours.

I. INTRODUCTION
II. FORMULATION
A. Review of Network Flow Theory
B. Clustering Rationale
C. The Clustering Algorithm and Its Properties
D. A Clustering Example
III. HIERARCHICAL IMPLEMENTATION
A. Extensions on Network Flow Theory
B. A Clustering Algorithm Using Hierarchical Implementation
C. Incorporation of Constraints
IV. IMAGE SEGMENTATION BASED ON CLUSTERING
V. EXPERIMENTAL RESULTS
A. Edge Based Segmentation of a MR Image
B. Edge Detection of An Airport Image
VI. CONCLUSION
REFERENCES

Knowls

Knowl 1 — Optimal K-Subgraph Partitioning via Gomory-Hu Cut Equivalent Trees
theoretical result
For an undirected data adjacency graph $G = (V, A)$ where each vertex represents a data point and each edge $(v_i, v_j) e ext{null}$ has a non-negative capacity $c_{ij}$ representing similarity, an unconstrained optimal $K$ -partition of $G$ into $K$ mutually exclusive subgraphs that minimizes the maximum inter-subgraph flow is obtained from a Gomory-Hu cut equivalent tree $T^*$ of $G$ .
1. Partition Optimality: Removing the $K-1$ arcs with the smallest capacities from $T^*$ partitions $V$ into $K$ connected components such that the maximum inter-subgraph maximum flow between any pair of distinct components is minimized among all possible $K$ -partitions of $G$ .
2. Within-Cluster Dominance: In this optimal partition, the maximum flow between any pair of vertices belonging to the same subgraph (intra-subgraph flow) is strictly greater than or equal to the maximum flow between vertices in distinct subgraphs (inter-subgraph flow).
3. Uniqueness: If $F_T$ is the smallest capacity among all remaining (unmarked) arcs in $T^*$ , and all $K-1$ removed arcs have capacities strictly less than $F_T$ , then this optimal $K$ -partition is strictly unique.
Knowl 2 — Subgraph Condensation Theorem for Maximum Flow Computation
theoretical result
Let $G = (V, A)$ be an undirected connected graph with positive arc capacities, and let $G_o = (V_o, A_o)$ be an arbitrary subgraph of $G$ . Let $F'_{o,\min} = \min_{u, v \in V_o} F'_{uv}$ be the minimum value of maximum flows between all pairs of vertices in $V_o$ computed using only the internal edges of subgraph $G_o$ . Let $G_c$ denote the condensed graph obtained from $G$ by collapsing all vertices in $V_o$ into a single vertex $v_o^c$ , summing parallel arc capacities.

For any two vertices $s, t \in V$ , if the true maximum flow $F_{st}$ in $G$ satisfies $F_{st} < F'_{o,\min}$ , then the maximum flow and the minimum cut between $s$ and $t$ in $G$ are identical to those computed between $s$ and $t$ in $G_c$ . In particular, the minimum cut separating $s$ and $t$ in $G$ cannot contain any internal arc of $G_o$ .
Knowl 3 — Partially Flow and Cut Equivalent Tree Construction
theoretical result
Let $G_o$ be a subgraph of an undirected graph $G$ , and let $F'_{o,\min} = \min_{u, v \in V_o} F'_{uv}$ be the minimum maximum flow across all pairs of vertices in $G_o$ computed locally within $G_o$ . Let $G_c$ be formed by condensing all vertices of $G_o$ into a single vertex $v_o^c$ , and let $T_c^*$ be a Gomory-Hu equivalent tree constructed on $G_c$ .

Then $T_c^*$ is a partially cut and flow equivalent tree of $G$ at threshold $F'_{o,\min}$ :
1. For any pair of vertices $s, t \in V$ with true maximum flow $F_{st} < F'_{o,\min}$ , the maximum flow $F_{st}$ in $G$ equals the minimum arc capacity on the unique path between $s$ and $t$ in $T_c^*$ .
2. For any pair of vertices $s, t \in V$ with $F_{st} \ge F'_{o,\min}$ , the flow value computed in $T_c^*$ is bounded below by $F'_{o,\min}$ (i.e., at least as large as $F_{st}$ ), ensuring that no arc with capacity below $F'_{o,\min}$ will be incorrectly cut.
Knowl 4 — Interior Branch Condensation of Subgraph Cut Trees
theoretical result
Let $G_o \subset G$ be a subgraph of an undirected graph $G$ . A vertex in $G_o$ is an interior vertex if all of its neighboring vertices in $G$ belong to $G_o$ ; otherwise it is a boundary vertex. Let $T_o^*$ be a Gomory-Hu equivalent tree constructed on $G_o$ .

Let $T_o^b$ be a branch (connected subtree) of $T_o^*$ consisting exclusively of interior vertices of $G_o$ , and let $T_r^b \in T_o^b$ be the root vertex of this branch that connects to a non-interior vertex in $T_o^*$ . Let $G_c$ be the graph obtained from $G$ by condensing all vertices of $T_o^b$ into a single vertex $v_o^b$ .

The full equivalent tree $T^*$ of $G$ can be constructed by:
1. Constructing an equivalent tree $T_c^*$ for the condensed graph $G_c$ .
2. Replacing the condensed vertex $v_o^b$ in $T_c^*$ with the branch $T_o^b$ , connecting $T_o^b$ to $T_c^*$ through its root vertex $T_r^b$ .
All minimum cuts inside $T_o^b$ are exact minimum cuts in $G$ and do not need to be recomputed.
Knowl 5 — Hierarchical Clustering Algorithm via Subgraph Condensation
algorithm
This algorithm constructs a partially equivalent cut tree $T_c^*$ for a large graph $G = (V, A)$ using multi-level hierarchical condensation and partitions the graph into clusters based on a capacity threshold $F_T$ .
Input: Undirected graph $G = (V, A)$ with non-negative arc capacities $c_{ij}$ , flow threshold $F_T$ .
Output: A set of disjoint subgraphs (clusters) partitioning $V$ .
1. Partition $G$ into small subgraphs of comparable size, denoted by the set $\{G_{0,m}\}$ . Set hierarchy level $i \leftarrow 0$ .
2. For each subgraph $G_{i,m}$ , compute its Gomory-Hu equivalent tree $T_{i,m}^*$ .
3. In each tree $T_{i,m}^*$ :
   a. Permanently condense all subtrees connected by arcs with capacity $\ge F_T$ .
   b. Temporarily condense all maximal interior branches to single vertices, yielding condensed subgraphs $\{G_{i,m,c}\}$ .
4. If $\{G_{i,m,c}\}$ contains only a single condensed subgraph:
   a. Set $T_c^*$ to its equivalent tree.
   b. Expand all temporarily condensed interior vertices in $T_c^*$ .
   c. Proceed to Step 6.
5. Else:
   a. Group spatially adjacent condensed subgraphs in $\{G_{i,m,c}\}$ into subsets.
   b. Form new combined subgraphs $\{G_{i+1,m}\}$ by restoring inter-subgraph arcs between the grouped components.
   c. Set $i \leftarrow i + 1$ and return to Step 2.
6. Remove all arcs in $T_c^*$ with capacities $< F_T$ in order of increasing capacity.
7. Return the resulting connected components of $T_c^*$ as the output clusters.
Knowl 6 — Local Edge Strength Computation on Pixel Lattices
equation
In a 4-connected 2D image grid where $x_{i,j}$ denotes the scalar gray-scale intensity at pixel coordinate $(i, j)$ , edge elements represent boundaries between adjacent pixels. The horizontal edge strength $D^H_{i,j}$ between pixel $(i, j)$ and $(i, j+1)$ , and the vertical edge strength $D^V_{i,j}$ between pixel $(i, j)$ and $(i+1, j)$ , are computed using directional derivative masks:

$D^H_{i,j} = \frac{\left| \delta (x_{i,j} - x_{i,j+1}) + (x_{i-1,j} - x_{i-1,j+1}) + (x_{i+1,j} - x_{i+1,j+1}) + (x_{i,j-1} - x_{i,j+2}) \right|}{(8 + 3\delta)\sigma}$

$D^V_{i,j} = \frac{\left| \delta (x_{i,j} - x_{i+1,j}) + (x_{i,j-1} - x_{i+1,j-1}) + (x_{i,j+1} - x_{i+1,j+1}) + (x_{i-1,j} - x_{i+2,j}) \right|}{(8 + 3\delta)\sigma}$

where $\delta \ge 0$ is a control parameter governing mask edge smoothing (set to $\delta = 7$ for brain MRI to reduce noise sensitivity while preventing thick edges), and $\sigma > 0$ is a scaling parameter proportional to the minimum intensity difference indicating a potential boundary.
Knowl 7 — Arc Capacity Assignment Function for Image Graphs
model/method
In graph-based image segmentation, pixels are represented as vertices and adjacent pixels are joined by arcs. To map edge strength $D^p_{i,j}$ (where $p \in \{H, V\}$ denotes horizontal or vertical edges) to arc capacity $C^p_{i,j}$ , a piecewise exponential function is employed:

$C^p_{i,j} = \begin{cases} e^{-(D^p_{i,j})^2}, & \text{if } D^p_{i,j} < 3 \\ e^{-3 D^p_{i,j}}, & \text{if } D^p_{i,j} \ge 3 \end{cases}$

This mapping assigns high capacities to weak edges (similar neighboring pixels) and low capacities to strong edges (dissimilar neighboring pixels). For $D^p_{i,j} < 3$ , a Gaussian-like quadratic decay provides sharp discrimination between weak and moderately strong edges. For $D^p_{i,j} \ge 3$ , the function switches to a slower linear-exponential decay $e^{-3 D^p_{i,j}}$ to prevent capacities from underflowing to numerical zero (e.g., below $10^{-12}$ ), avoiding the creation of tiny, isolated false clusters during cut removal.
Knowl 8 — Small-Cluster Merging and Unclassified Pixel Labeling
model/method
To handle over-segmentation resulting from capacity thresholding at $F_T$ , a post-clustering refinement procedure is applied:
1. Clusters containing fewer than 5 pixels are designated as candidate small clusters.
2. Unless a small cluster is separated in the cut tree $T_c^*$ by arc capacities substantially smaller than $F_T$ , it is merged into the spatially adjacent neighboring cluster with $\ge 5$ pixels that has the smallest absolute difference in mean pixel intensity.
3. Any small cluster ( $< 5$ pixels) that cannot be merged into an existing neighbor is marked as "unclassified" to prevent noisy or ambiguous boundary pixels from distorting region statistics.
Knowl 9 — Brain MRI Tissue Segmentation Performance
empirical result
The hierarchical graph-theoretic clustering algorithm was evaluated on a $256 \times 256$ cross-sectional brain magnetic resonance (MR) image of a patient with white matter lesions using parameters $\sigma = 3.0$ , $\delta = 7.0$ , and flow threshold $F_T = 0.1$ :
- Thresholding at $F_T = 0.1$ followed by small-cluster merging ( $< 5$ pixels) segmented the image into 161 total clusters.
- Pruning the skull and external tissues using the equivalent tree isolated 79 brain tissue clusters.
- Interactive labeling grouped the 79 clusters into four primary clinical tissue categories: grey matter, white matter, ventricles, and tumor.
- Unlike the Marr-Hildreth zero-crossing operator (which merged pixels from white matter and ventricles into single false regions due to shared zero crossings), the graph cut approach accurately maintained closed contours without forming spurious connections between distinct anatomical structures.
Knowl 10 — Computational Acceleration of Hierarchical Graph Condensation
empirical result
On a 4-connected $256 \times 256$ pixel image graph containing 65,536 vertices:
- Direct application of the standard Gomory-Hu algorithm on a Sun SPARCstation requires solving 65,535 maximum flow problems on full or near-full graphs, requiring over 12 hours of CPU time.
- The hierarchical condensation algorithm (employing a three-level hierarchy to condense subgraphs and eliminate interior branches) reduced total CPU time to approximately 10 minutes on the same hardware.
- For all cuts with capacity below threshold $F_T$ , the resulting partially equivalent tree is identical to the exact Gomory-Hu cut tree of the full graph, guaranteeing identical global optimality.

Coverage note — A simple synthetic 12-patch illustrative example (Section II-D) and an aerial photograph edge detection experiment (Section V-B) were omitted in favor of the core theoretical condensation theorems, the primary hierarchical clustering algorithm, and the detailed brain MRI evaluation.

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Citation

MLA

Wu, Z., and R. Leahy. “An Optimal Graph Theoretic Approach to Data Clustering: Theory and Its Application to Image Segmentation”. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 11, 1993, pp. 1101–13, https://doi.org/10.1109/34.244673.

APA

Wu, Z., & Leahy, R. (1993). An optimal graph theoretic approach to data clustering: theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(11), 1101–1113. https://doi.org/10.1109/34.244673

Chicago

Wu, Z., and R. Leahy. 1993. “An Optimal Graph Theoretic Approach to Data Clustering: Theory and Its Application to Image Segmentation”. IEEE Transactions on Pattern Analysis and Machine Intelligence 15 (11): 1101–13. https://doi.org/10.1109/34.244673.

Harvard

Wu, Z. and Leahy, R. (1993) “An optimal graph theoretic approach to data clustering: theory and its application to image segmentation”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(11), pp. 1101–1113. Available at: https://doi.org/10.1109/34.244673.

Vancouver

1. Wu Z, Leahy R (1993) An optimal graph theoretic approach to data clustering: theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15:1101–1113

BibTeX

@article{Wu_1993, title={An optimal graph theoretic approach to data clustering: theory and its application to image segmentation}, volume={15}, ISSN={0162-8828}, url={http://dx.doi.org/10.1109/34.244673}, DOI={10.1109/34.244673}, number={11}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Wu, Z. and Leahy, R.}, year={1993}, pages={1101–1113} }

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An Optimal Graph Theoretic Approach to Data Clustering: Theory and Its Application to Image Segmentation

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Table of Contents

Knowls

Knowl 1 — Optimal K-Subgraph Partitioning via Gomory-Hu Cut Equivalent Trees

Knowl 2 — Subgraph Condensation Theorem for Maximum Flow Computation

Knowl 3 — Partially Flow and Cut Equivalent Tree Construction

Knowl 4 — Interior Branch Condensation of Subgraph Cut Trees

Knowl 5 — Hierarchical Clustering Algorithm via Subgraph Condensation

Knowl 6 — Local Edge Strength Computation on Pixel Lattices

Knowl 7 — Arc Capacity Assignment Function for Image Graphs

Knowl 8 — Small-Cluster Merging and Unclassified Pixel Labeling

Knowl 9 — Brain MRI Tissue Segmentation Performance

Knowl 10 — Computational Acceleration of Hierarchical Graph Condensation

References

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