In this article, an implementation of an efficient graphbased image segmentation technique will be described, this algorithm was proposed by Felzenszwalb et. al. from MIT. The slides on this paper can be found from Stanford Vision Lab.. The algorithm is closely related to Kruskal’s algorithm for constructing a minimum spanning tree of a graph, as stated by the author and hence can be
implemented to run in O(m log m) time, where m is the number of edges in the graph.
In the case of image segmentation, the elements in V are pixels and the weight of an edge is some measure of the dissimilarity between the two pixels connected by that edge (e.g., the difference in intensity, color, motion, location or some other local attribute).
Particularly for the implementation described here, an edge weight functionbased on the absolute intensity difference (in the yiq space) between the pixels connected by an edge, w((vi, vj )) = I(pi) − I(pj ).
In the graphbased approach, a segmentation S is a partition of V into components
such that each component (or region) C ∈ S corresponds to a connected component
in a graph G0 = (V, E0), where E0 ⊆ E.
In other words, any segmentation is induced by a subset of the edges in E. There are different ways to measure the quality of a segmentation but in general we want the elements in a component to be similar, and elements in different components to be dissimilar.
This means that edges between two vertices in the same component should have relatively low weights, and edges between vertices in different components should have higher weights.
The next figure shows the steps in the algorithm. The algorithm is very similar to Kruskal’s algorithm for computing the MST for an undirected graph.
The threshold function τ controls the degree to which the difference between two
components must be greater than their internal differences in order for there to be
evidence of a boundary between them.
For small components, Int(C) is not a good estimate of the local characteristics of the data. In the extreme case, when C = 1, Int(C) = 0. Therefore, a threshold function based on the size of the component, τ (C) = k/C is needed to be used, where C denotes the size of C, and k is some constant parameter.
That is, for small components we require stronger evidence for a boundary. In practice k sets a scale of observation, in that a larger k causes a preference for larger components.
In general, a Gaussian filter is used to smooth the image slightly before computing the edge weights, in order to compensate for digitization artifacts. We always use a Gaussian with σ = 0.8, which does not produce any visible change to the image but helps remove artifacts.
The following python code shows how to create the graph.
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import numpy as np from scipy import signal import matplotlib.image as mpimg def gaussian_kernel(k, s = 0.5 ): # generate a (2k+1)x(2k+1) gaussian kernel with mean=0 and sigma = s probs = [exp(  z * z / ( 2 * s * s)) / sqrt( 2 * pi * s * s) for z in range (  k,k + 1 )] return np.outer(probs, probs) def create_graph(imfile, k = 1. , sigma = 0.8 , sz = 1 ): # create the pixel graph with edge weights as dissimilarities rgb = mpimg.imread(imfile)[:,:,: 3 ] gauss_kernel = gaussian_kernel(sz, sigma) for i in range ( 3 ): rgb[:,:,i] = signal.convolve2d(rgb[:,:,i], gauss_kernel, boundary = 'symm' , mode = 'same' ) yuv = rgb2yiq(rgb) (w, h) = yuv.shape[: 2 ] edges = {} for i in range (yuv.shape[ 0 ]): for j in range (yuv.shape[ 1 ]): #compute edge weight for nbd pixel nodes for the node i,j for i1 in range (i  1 , i + 2 ): for j1 in range (j  1 , j + 2 ): if i1 = = i and j1 = = j: continue if i1 > = 0 and i1 = 0 and j1 < h: wt = np. abs (yuv[i,j, 0 ]  yuv[i1,j1, 0 ]) n1, n2 = ij2id(i,j,w,h), ij2id(i1,j1,w,h) edges[n1, n2] = edges[n2, n1] = wt return edges 
Output Images for two different values of the parameter k
Output Images for two different values of the parameter k
Output Segmented Images
Output Images for two different values of the parameter k
Segmented Output Image
Posted 10 May 2021
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