Computer Graphics and Image Processing (CIS-601)

Image Segmentation by Clustering (Using Mahalanobis Distance) - Manjit Chintapalli

What is Image Segmentation ? To humans, an image is not just a random collection of pixels; it is a meaningful arrangement of regions and objects. There also exits a variety of images: natural scenes, paintings, etc. Despite the large variations of these images, humans have no problem to interpret them.

Image Segmentation ? (Contd..) Image segmentation is the first step in image analysis and pattern recognition. It is a critical and essential component of image analysis system, is one of the most difficult tasks in image processing, and determines the quality of the final result of analysis.

The Process… Image segmentation is the process of dividing an image into different regions such that each region is homogeneous.

Image Segmentation methods

Image segmentation methods can be categorized as follows (this is not an exhaustive list): Histogram thresholding Edge-based approaches Region-based approaches Hybrid: consider both edges and regions.

Continued… Segmentation of texture by PCA Segmentation of texture with moments

Color Image Segmentation Image segmentation based on color Image segmentation based on color We use clustering to segment an image according to color features We use clustering to segment an image according to color features

Clustering ? Clustering is a common step to get a segmentation In the data space, clusters are regarded as regions of similar data points

Clustering…. Similar data points grouped together into clusters.

Clustering… Most popular clustering algorithms suffer from two major drawbacks First, the number of clusters is predefined, which makes them inadequate for batch processing of huge image databases Secondly, the clusters are represented by their centroid and built using an Euclidean distance therefore inducing generally an hyperspheric cluster shape, which makes them unable to capture the real structure of the data. This is especially true in the case of color clustering where clusters are arbitrarily shaped

Clustering Algorithms K-means K-means K-medoids K-medoids Hierarchical Clustering Hierarchical Clustering There are many other algorithms used for clustering. There are many other algorithms used for clustering.

K-means Clustering Algorithm Step 1: Choose K cluster centers: c¹(1),c²(1),c³(1)….. Step 2: At the kth iterative step distribute the samples among the K cluster domains, using the relation for all x in S(j), if |x – Cj (k)| < |x – Cj (k)| Step 3: Compute the new cluster centers

Algorithm continued … Step 4: If the algorithm has converged and the procedure is terminated. Otherwise go to Step 2 S(j): contains data points in RGB space x: is the data point in an iteration C(j): is the center of the cluster

How the problem was approached ? First the images are read from the directory. First the images are read from the directory. Then each image is transformed into a 3-D RGB space. Then each image is transformed into a 3-D RGB space. K-means clustering using Mahanalobis distance and Euclidean distance was applied. K-means clustering using Mahanalobis distance and Euclidean distance was applied.

Continued….. Next, the mean color of each cluster is calculated. Next, the mean color of each cluster is calculated. Then it is transformed back from 3-D space. Then it is transformed back from 3-D space. Result : Segmented image Result : Segmented image

Mahalanobis Distance M.D. is a very useful way of determining the ”similarity” of a set of values from an ”unknown”: sample to a set of values measured from a collection of ”known” samples M.D. is a very useful way of determining the ”similarity” of a set of values from an ”unknown”: sample to a set of values measured from a collection of ”known” samples Superior to Euclidean distance because it takes distribution of the points (correlations) into account Superior to Euclidean distance because it takes distribution of the points (correlations) into account Traditionally to classify observations into different groups Traditionally to classify observations into different groups

Mahalanobis vs. other classical statistical approaches It takes into account not only the average value but also its variance and the covariance of the variables measured It takes into account not only the average value but also its variance and the covariance of the variables measured It compensates for interactions (covariance) between variables It compensates for interactions (covariance) between variables It is dimensionless It is dimensionless

Mahalanobis Distance D t (x) = (x – m t ) S -1 t (x – m t )` D t (x) = (x – m t ) S -1 t (x – m t )` Mahalanobis Distance

Euclidean Distance The Euclidean distance is the straight- line distance between two pixels The Euclidean distance is the straight- line distance between two pixels Euc dist = √((x1 - x2)² + (y1 - y2)²), where (x1,y1) & (x2,y2) are two pixel points or two data points Euc dist = √((x1 - x2)² + (y1 - y2)²), where (x1,y1) & (x2,y2) are two pixel points or two data points

After executing the program… Original Images

More images…

During execution… The images are transformed into 3-D RGB space The images are transformed into 3-D RGB space K-means clustering is applied which groups data points with similar color into clusters K-means clustering is applied which groups data points with similar color into clusters Example:

Continued… The data points with minimum mahalanobis distance are grouped together into clusters. The data points with minimum mahalanobis distance are grouped together into clusters. After grouping into clusters, the mean color of the cluster is taken and mapped back into the image After grouping into clusters, the mean color of the cluster is taken and mapped back into the image

The program was executed with different number of clusters The program was executed with different number of clusters The same program was run with Euclidean distance for distance between data points The same program was run with Euclidean distance for distance between data points Continued…

Segmented Images

Segmented with 4 clusters Original Image Segmented with 6 clusters

Original Image 2 clusters 4 Clusters 6 Clusters

Original Image 4 Clusters 2 Clusters 6 Clusters

Mahalanobis vs. Euclidean

4 clusters Mahalanobis 4 clusters Euclidean Original Image

6 Clusters Mahalanobis 6 Clusters Euclidean More detailed description using Mahalanobis

Original Image 6 Clusters Mahalanobis 6 Clusters Euclidean Image more clear than Euclidean

Mahalanobis vs. Euclidean Result : Using Mahalanobis distance in the k- means clustering proved to be better than Euclidean distance. Using Mahalanobis distance in the k- means clustering proved to be better than Euclidean distance. This is because the Mahalanobis considers the covariance between variables. This is because the Mahalanobis considers the covariance between variables.

Conclusion The program runs with all jpeg images and takes a lot of time to run for larger images. The program runs with all jpeg images and takes a lot of time to run for larger images. This is due to the calculation of the inverse of the covariance matrix for the Mahalanobis distance. This is due to the calculation of the inverse of the covariance matrix for the Mahalanobis distance. The results show that Image segmentation using Mahalanobis distance works and is better than Euclidean distance. The results show that Image segmentation using Mahalanobis distance works and is better than Euclidean distance.

The End