Minimization problem image processing

Hello,

I'm trying to minimize this function (by p):

where:

p is a NxN image

k is NxN image

tc is NxN image

* is the convolution of image k and image p

grad() is the gradient of the image

||·||_L1 is the L1 norm

||·||_L2 is the L2 norm

theta is a constant

I need a little bit of help coding this one in vector form.

[OJ27]

I have looked at some implementations without the convolution, but this part is just throwing me off. I have read Young's inequality but don't know how to implement it.