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I have a 2-D discrete signal in which each point can be represented as $(x, y)$. These points are varying with time $t$. Can we represent the TV-norm using the following formula?

$$\sum_{t}|x_t - x_{t-1}| + |y_t - y_{t-1}|$$

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  • $\begingroup$ What is the/your definition of the TV-norm? $\endgroup$ – user883521 Dec 7 '17 at 19:04
  • $\begingroup$ Sorry, could not get you. Can you please elaborate? $\endgroup$ – Avijit Dasgupta Dec 7 '17 at 19:31
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Your definition looks fine to me, if your goal is to treat the signal at every fixed $(x,y)$ location as a function of time and then compute 1D total variation along time. Is that what you intend to do?

Here are a couple of other ways you can define it (again, I have no clue what your application is, and why you want to compute the TV norm in the first place, so I don't know which of these make sense in your application).

To fix notation, let's denote your signal of interest as $z(n_x,n_y,n_t)$ where $z$ is the quantity measured at discrete locations indexed by $(n_x,n_y)$ in space and at a time sample number $n_t$.

TV norm for each fixed spatial location: Fix a location $(n_x^*,n_y^*)$. $$ TV = \sum_{n_t} |z(n_x^*,n_y^*,n_t+1) - z(n_x^*,n_y^*,n_t)| $$

TV norm for each 2D time snapshot: Fix a time instant $n^*_t$. $$ TV = \sum_{n_x,n_y} |z(n_x+1,n_y,n^*_t) - z(n_x,n_y,n^*_t)| + | z(n_x,n_y+1,n^*_t) - z(n_x,n_y,n^*_t)| $$

TV norm in 3D space-time:

$$ TV = \sum_{n_x,n_y,n_t} |z(n_x+1,n_y,n_t) - z(n_x,n_y,n_t)| + | z(n_x,n_y+1,n_t) - z(n_x,n_y,n_t)| + |z(n_x,n_y,n_t+1) - z(n_x,n_y,n_t)| $$

See also: https://en.wikipedia.org/wiki/Total_variation_denoising#2D_signal_images

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  • $\begingroup$ I would like to reduce the variation of the signal in the time dimension. So, the first definition should work, right? $\endgroup$ – Avijit Dasgupta Dec 7 '17 at 19:33
  • $\begingroup$ Yes, the first one should work. $\endgroup$ – Atul Ingle Dec 7 '17 at 21:15

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