I am working on an optimization-based image denoising project in which I have three "flavors" of an optimization problem, one constrained and two unconstrained. They are given as follows:

$$\min_{\mathbf{x} \in \mathcal{R}^{MN}} \|\mathbf{x}-\mathbf{y}\|_2^2 + \lambda \|{Ax}\|_2^2 \tag{1}$$

$$\min_{\mathbf{x} \in \mathcal{R}^{MN}} \|\mathbf{x}-\mathbf{y}\|_2^2 \;\;\; \mbox{subject to} \;\;\; \|\mathbf{Ax}\|_2 < \alpha \tag{2}$$

$$\min_{\mathbf{x} \in \mathcal{R}^{MN}} \|\mathbf{Ax}\|_2^2 \;\;\; \mbox{subject to} \;\;\; \|\mathbf{x}-\mathbf{y}\|_2 < \beta \tag{3}$$

where $\mathbf{y}$ is the noisy image (with Gaussian noise with known standard deviation $\sigma$) and $\mathbf{x}$ is the denoised image (both in column-stacked form). Moreover, $\mathbf{A} = [\mathbf{D_x}^T \: \mathbf{D_y}^T]^T$ (a transformation which returns the image gradient values in one $MN \times 1$ vector, where $M$ is the number of rows and $N$ is the number of columns in the image). I'm trying to understand the intuition behind the two constraints. In general, what I see is that the we are trading off "closeness to the noisy image" ($\|\mathbf{x}-\mathbf{y}\|_2$) and blurriness ($\|\mathbf{Ax}\|_2$). In the first constrained problem, $\alpha$ reflects the amount of blurriness we allow in the denoised image. According to my instructor, $\beta$ is related to to the noise standard deviation in that $\beta = \sigma\sqrt{n}$, where $n$ is the total number of pixels in the image. This is where I'm really confused. What does noise standard deviation have to do with "closeness to the noisy image" which is my interpretation of $\|\mathbf{x}-\mathbf{y}\|_2$? Do I have to set $\beta$ to $\sigma\sqrt{n}$ or can I choose it to be whatever I want it to be like I can for $\alpha$ in the first constrained problem? For example, if I have a desired final noise standard deviation in mind, how do I set $\beta$ accordingly?


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