Say I have a complex function $f$ that has a near piece-wise constant magnitude, but a non constant phase.

If I have an objective function with a total variation term (e.g. for denoising or compressed sensing) it usually has the following form:

$$ obj_1(f) = \ldots + \text{TV}(f) $$

However, since I assume that $f$ has a piece-wise constant magnitude, I think it might be better to use:

$$ obj_2(f) = \ldots + \text{TV}(|f|) $$

However, for a gradient based solver, one would have to know the gradient of obj2. The gradient for $obj_1(f)$ is: $\text{TV}'\left(TV(f)\right)$. What is the gradient of $obj_2(f)$?

Update:

Intuitively I would assume something like the following (since the phase has no influence on $obj_2$, leave the phase "untouched"):

$$ \text{TV}'\left(TV(|f|)\right)* e^{i \arg(f)} $$

However, my knowledge in complex analysis is very limited and I am not sure if this makes sense.

Say I have a complex function $f^*$ (e.g. a MRI image) that has a near piece-wise constant magnitude, but a non constant phase.

If I have an optimization problem to find $f^*$ and set up an objective function with a total variation term (e.g. for denoising or compressed sensing) it usually has the following form:

$$ obj_1(f) = \ldots + \text{TV}(f) $$

However, since I assume that $f$ has a piece-wise constant magnitude, I think it might be better to use:

$$ obj_2(f) = \ldots + \text{TV}(|f|) $$

However, for a gradient based solver, one would have to know the gradient of obj2. The gradient for $obj_1(f)$ is: $\text{TV}'\left(TV(f)\right)$. What is the gradient of $obj_2(f)$?

Update:

Intuitively I would assume something like the following (since the phase has no influence on $obj_2$, leave the phase "untouched"):

$$ \text{TV}'\left(TV(|f|)\right)* e^{i \arg(f)} $$

However, my knowledge in complex analysis is very limited and I am not sure if this makes sense.

Say I have a complex function $f$ that has a near piece-wise constant magnitude, but a non constant phase.

If I have an objective function with a total variation term (e.g. for denoising or compressed sensing) it usually has the following form:

$$ obj_1(f) = \ldots + \text{TV}(f) $$

However, since I assume that $f$ has a piece-wise constant magnitude, I think it might be better to use:

$$ obj_2(f) = \ldots + \text{TV}(|f|) $$

However, for a gradient based solver, one would have to know the gradient of obj2. The gradient for $obj_1(f)$ is: $\text{TV}'\left(TV(f)\right)$. What is the gradient of $obj_2(f)$?

-- Update:

Intuitively I would suggest something like this, since the phase has no influence on the objective function, I would leave the phase "untouched":

$$ \text{TV}'\left(TV(|f|)\right)* e^{i \arg(f)} $$

However, my knowledge in complex analysis is very limited and I am not sure if this makes sense.

Say I have a complex function $f$ that has a near piece-wise constant magnitude, but a non constant phase.

If I have an objective function with a total variation term (e.g. for denoising or compressed sensing) it usually has the following form:

$$ obj_1(f) = \ldots + \text{TV}(f) $$

However, since I assume that $f$ has a piece-wise constant magnitude, I think it might be better to use:

$$ obj_2(f) = \ldots + \text{TV}(|f|) $$

However, for a gradient based solver, one would have to know the gradient of obj2. The gradient for $obj_1(f)$ is: $\text{TV}'\left(TV(f)\right)$. What is the gradient of $obj_2(f)$?

Update:

Intuitively I would assume something like the following (since the phase has no influence on $obj_2$, leave the phase "untouched"):

$$ \text{TV}'\left(TV(|f|)\right)* e^{i \arg(f)} $$

However, my knowledge in complex analysis is very limited and I am not sure if this makes sense.