# Image Segmentation using Graph Cut Methods: Exercise

## Exercise Overview

This exercise focuses on **Graph Cut** methods for image segmentation, building upon your mathematical background in eigenvalues and eigenvectors. Graph Cut formulates image segmentation as a graph partitioning problem that can be solved using spectral methods.

## Theoretical Background

### Graph Representation of Images

In Graph Cut segmentation, we represent an image as an undirected graph $G = (V, E)$ where:

- **Vertices (V)**: Pixels in the image ($v_i \in V$)
- **Edges (E)**: Connections between neighboring pixels ($e_{ij} \in E$)
- **Edge Weights ($w_{ij}$)**: Similarity between pixels, typically defined as:

$$ w_{ij} = \exp\left(-\frac{\|I(i) - I(j)\|^2}{\sigma^2}\right) $$

where $I(i)$ and $I(j)$ are pixel intensities, and $\sigma$ is a scaling parameter.

### Graph Cut Formulation

The segmentation problem is formulated as partitioning the graph into two disjoint sets $A$ and $B$ (foreground and background) by minimizing a cost function:

$$ \text{Cut}(A, B) = \sum_{i \in A, j \in B} w_{ij} $$

However, the simple cut criterion tends to favor small, isolated components. This leads to the **Normalized Cut** criterion:

$$ \text{NCut}(A, B) = \frac{\text{Cut}(A, B)}{\text{Assoc}(A, V)} + \frac{\text{Cut}(A, B)}{\text{Assoc}(B, V)} $$

where $\text{Assoc}(A, V) = \sum_{i \in A, j \in V} w_{ij}$ is the total connection from set $A$ to all vertices.

### Spectral Solution

The Normalized Cut problem can be solved using spectral graph theory. We define:

- **Degree matrix** $D$: Diagonal matrix with $D_{ii} = \sum_j w_{ij}$
- **Laplacian matrix** $L = D - W$, where $W$ is the weight matrix

The solution involves finding the eigenvector corresponding to the second smallest eigenvalue of the generalized eigenvalue problem:

$$ (D - W)\mathbf{y} = \lambda D\mathbf{y} $$

The signs of the elements in this eigenvector provide the segmentation.

## Exercise Tasks

### Part 1: Graph Construction and Matrix Formulation

1. **Build the Weight Matrix**
   - Implement a function to construct the affinity matrix $W$ for a given image
   - Use both intensity and spatial proximity to compute weights
   - Experiment with different neighborhood systems (4-connected vs 8-connected)

2. **Matrix Operations**
   - Compute the degree matrix $D$
   - Construct the Laplacian matrix $L = D - W$
   - Verify properties of the Laplacian matrix

### Part 2: Spectral Segmentation

- Solve the generalized eigenvalue problem $(D - W)\mathbf{y} = \lambda D\mathbf{y}$
- Extract the eigenvector corresponding to the second smallest eigenvalue
- Use this eigenvector to bipartition the image

## Implementation Guidelines

### Key Mathematical Components:

1. **Affinity Matrix Construction**: 
   - Size: $N \times N$ where $N$ is the number of pixels
   - Sparse implementation recommended for efficiency

2. **Eigenvalue Computation**:
   - Use specialized algorithms for large sparse matrices
   - Focus on finding the smallest eigenvalues/eigenvectors

3. **Segmentation from Eigenvectors**:
   - Threshold the Fiedler vector (second smallest eigenvector)
   - Use k-means on multiple eigenvectors for multi-way cuts

## Deliverables

Submit your Jupyter Notebook in FUM-VU.

## References

For deeper understanding of Graph Cut methods and their applications, please refer to:
{cite}`Fayaz2015Spectral, Hoseini2014graph`

These references provide comprehensive coverage of graph theory and its applications in image segmentation & graph cut, connecting the mathematical concepts with practical implementations.

This exercise will help you bridge the gap between abstract mathematical concepts (eigenvalues, eigenvectors, matrix theory) and practical image analysis applications, demonstrating the power of linear algebra in solving complex computer vision problems.