# 2D moving average model for image synthesis - how to interpret this algorithm?

Background: I'm working on a 2D Moving-Average (MA) estimator for image processing stuff and I'm trying to follow along from a paper that defines the algorithm for estimation.

The main equation for the parameter estimation is: $$\eta_i = (1/N_i) \sum_{u \in \Omega_i}|Y(u)|^2$$

where $$\Omega_i = \{u \in \Omega_u | ||Au|| = \rho_i\}$$

However, $$\Omega_u$$ is defined earlier as $$\Omega_u = \{(u_1, u_2) \in \Omega | 0 \le u_1 \lt N, 0 \le u_2 \le N/2 , (u_1, u_2) \neq (0,0)\}$$. Since we're dealing with images, $$N$$ is the image size, which in my case is 128 for a 128x128 image. $$u_1, u_2$$ are both indices in the frequency domain after the DFT is used on the original image.

I'm hung up on what the set $$\Omega_i$$ actually entails and consequently, what the I'm summing over. Is this saying there are two sets like $$\Omega_{i1}$$ and $$\Omega_{i2}$$?

My current interpretation is that $$u_1 = \{0...127\}, u_2 = \{0...64\}$$ where $$u_1$$ is the "row" of $$Y$$ and $$u_2$$ is the "column" of $$Y$$. Then in the algorithm, I'm summing across the row of $$Y$$ for all $$u_2$$'s. Yet this leads to drastically different results for $$\eta_i$$ than what the paper has.

• Could you share a link to the paper?
– Royi
Commented May 11, 2023 at 12:05
• $\Omega_i$ is related to $\rho_i$ via $A$: it includes the frequency points such that their image by the linear transformation $A$ has the given norm. This corresponds to a pair of parallel lines. The number and values of the $\rho_i$ must be specified elsewhere.
– user67664
Commented May 11, 2023 at 12:31
• @Royi here is the link on IEEE: ieeexplore.ieee.org/document/730388 Commented May 11, 2023 at 18:16
• @YvesDaoust so $\Omega_I$ shouldn't be considered to be related to $\Omega_u$? I'm interpreting the set as containing "pairs" $u_1$, $u_2$. Also yes, $\rho_i$ is defined elsewhere as $\{\rho_i\} = \{||Au|| | u \in \Omega_u\}$. So it's just the reverse of what $\Omega_i$ is being defined as/ Commented May 11, 2023 at 18:18
• You did not supply information for the question to be answered then.
– user67664
Commented May 11, 2023 at 18:30