discrete signals - Using Total Variation Denoising to Clean Accelerometer Data

I know this is maybe a very basic question but I am doing this as a hobby and I can't find a solution to this problem. Basically I am trying to remove some noise from data I am reading from an accelerometer. This is what I want to achieve (Taken from Total Variation Denoising (An MM algorithm)):

I read in Picking the correct filter for accelerometer data that Total Variaton Denoising would fit my needs. So I read Wikipedia - Total Variation Denoising article from Wikipedia and I think I have to use one of this equations:

But I don't understand how I apply this to my signal. Suppose I have a set of x,y points like in the plots above, how I apply the equation to that data? I implemented some simple low-pass and high-pass filters like this:

gravity[0] = alpha * gravity[0] + (1 - alpha) * event.values[0];

But this is maybe too complex and I don't know where to start or how. I want to implement this in Java or C so Matlab is not an option (I have seen a lot of MatLab implementing this). I will appreciate any help to guide me in the right direction!

Answer

Apart from Total Variation Denoising you could try a first much simpler approach: a median-filter. You just move a window along your data and replace the current input value by the median of all data in the window. You just have to optimize the window length (by experimenting).

By the way, the equations you copied into your question are for 2-dimensional data, but your data are 1-D.