Recoloring pixels according to their cluster, we obtain a segmentation of the original image. Although mean shift is a reasonably versatile algorithm, it has primarily been applied to problems in computer vision, where it has been used for image segmentation, clustering, and video tracking. Application to big data problems can be challenging due to the fact the algorithm can become relatively slow in this limit.

A general formulation of the mean shift algorithm can be developed through consideration of density kernels. These effectively work by smearing out each point example in space over some small window. Summing up the mass from each of these smeared units gives an estimate for the probability density at every point in space by smearing, we are able to obtain estimates at locations that do not sit exactly atop any example.

This approach is often referred to as kernel density estimation — a method for density estimation that often converges more quickly than binning, or histogramming, and one that also nicely returns a continuous estimate for the density function. The first requirement is needed to ensure that our estimate is normalized, and the second is associated with the symmetry of our space. Two popular kernel functions that satisfy these conditions are given by. Below we plot an example in 1-d using the gaussian kernel to estimate the density of some population along the x-axis.

You can see that each sample point adds a small Gaussian to our estimate, centered about it: The equations above may look a bit intimidating, but the graphic here should clarify that the concept is pretty straightforward. Recall that the basic goal of the mean shift algorithm is to move particles in the direction of local increasing density. To obtain an estimate for this direction, a gradient is applied to the kernel density estimate discussed above. With this estimate, then, the mean shift algorithm protocol becomes.

As a final step, one determines which examples have ended up at the same points, marking them as members of the same cluster. Mean shift clustering Mean shift clustering is a general non-parametric cluster finding procedure — introduced by Fukunaga and Hostetler [ 1 ], and popular within the computer vision field.

Segmenting a color photo In the first example, we were using mean shift clustering to look for spatial clusters. Discussion Although mean shift is a reasonably versatile algorithm, it has primarily been applied to problems in computer vision, where it has been used for image segmentation, clustering, and video tracking. Mean shift pros: No assumptions on the shape or number of data clusters. The procedure only has one parameter, the bandwidth. Mean shift cons: Output does depend on bandwidth: too small and convergence is slow, too large and some clusters may be missed.

Computationally expensive for large feature spaces. Often slower than K-Means clustering Technical details follow. Technical background Kernel density estimation A general formulation of the mean shift algorithm can be developed through consideration of density kernels. Example of a kernel density estimation using a gaussian kernel for each data point: Adding up small Gaussians about each example returns our net estimate for the total density, the black curve. I have a question related to the first and second figures KDE surface and contour plot of this surface, respectively.

The mean shift clustering algorithm – EFavDB

How did you generate these figures with Matplotlib? Adam, do you have this Matlab code available anywhere? I am trying to generate like you did, the KDE surface plots and the contour plots of a dataset on which later I am going to apply your mean shift algorithm. I want to do that in order to have an idea of the expected number of clusters that mean shift will generate.

If I am not mistaken, your version of the algorithm uses a gaussian kernel.

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For bandwidth values larger than 1 or 2, I get a huge cone for the whole dataset. For bandwidth values larger than one, I get surface plots looking like a cone for the whole dataset, which theoritically would lead to one cluster. But, the mean shift algorithm generates different number of clusters. Have you also used the same functions for these plots?

Do you have an idea what might be going wrong? It says that the bandwidth is determined automatically unless you provide a scalar value. I imagine you are providing a scalar. I may take another look later if I have some time. Yes, I provide a scalar value. With the same value I feed the mean shift algorithm as well. Mmm, I see. What procedure did you follow for the generation of the KDE surface plot?

My data is 2D, as yours. According to what I understood KDEs are estimated probability distribution functions based on an assumed underlying distribution like guassian or epanechnikov distributions, is that right? In this example the KDE concept is used to conceptualize a probability function that could have generated the data that we have.

In other words, we start with some data 2D points. You can imagine that if we randomly sampled some probability distribution, we may have ended up with those points. What would such a probability function look like? Constructing a KDE helps us to answer this question. The KDE is constructed by placing a kernel weighting function, or conceptually a little hill or bump on each point. Adding all of those kernels up gives us the overall KDE function. Depending on the size of the kernel defined by the bandwidth value , the shape of the overall KDE surface will vary.

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I tried to make function of mean shift clustering on C. Thank you for your post, this was very helpful especially to a non-technical like me. I may have a basic question here, hope you can help. I ran your code and saw that your data. My question is, what if my data has three or more dimensions i. How do I set up the data? My implementation should work for data of any dimension.

• A review of mean-shift algorithms for clustering!
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Have you tried running it with higher dimensional data? Hi Matt, I also read your post, your post is helpful for me. As you know, in mean shift clustering, window-size is very important. If this value is small, computational time is very long, and this value is big, the quality of clustering is low.

Can I ask to you your good comment about window-size?? Hi Matt, Thanks for the blog!

Phase Classification by Mean Shift Clustering of Multispectral Materials Images

It is wonderfully written : I understand that the bandwidth value for the kernel is the most important parameter that decides the cluster means. You have said that the Gaussian kernel is the most commonly used one. But does the type of kernel we use influence the output? If we know that our clusters have a specific shape in space, then would it make sense to use a kernel function that approximates that shape?

However, Gaussian kernels should allow most shapes to be recognized in clustering results. One of the nice aspects of Mean Shift is that it is able to cluster arbitrary e.

The mean shift clustering algorithm

The kernel defines the shape of importance around each point. Gaussian kernels allow you to give equal importance in every direction from a given point, but as you get further away from the point the importance decays in an gaussian way. For example, you could use a flat also called uniform kernel, which usually defines a circular or spherical region around each point, and gives equal weight to all points within the region, and zero weight to points outside. Alternatively, you could use a linear kernel that allows the weight to decay in a linear manner as you get further away from each point.

Often times these kernels are used to improve performance, as you can ignore points that fall outside of the kernel during each iteration of the algorithm if you can figure out what point those are efficiently. The goal of the kernel is to help understand the probability function that may have generated the data. I have seen adaptive kernel approaches that vary the kernel bandwidth depending on the density of points in a local area, in an attempt to better estimate the underlying probability function. Hi Matt, Thanks for the reply! I am trying to extract ellipsoidal objects from lidar data and I have some promising first results from mean-shift.

Your answer helped me understand better what I am doing. And I feel that Gaussian kernel is a better choice compared to the ones available. Have a nice day! Hi Matt, Thanks for your information. As I understood, mean shift algorithm needs input value of band width like K value in k-means algorithm. If input K value is disadvantage in k-means then input band width is also one of the disadvantage. If so, how to overcome this? And also, how to overcome is its slowness?