Analysis and Synthesis

The last lesson and a new frame

Three sampling regimes

Self-inverting transforms

A map of what follows

A transform analyzes a 256-pixel image into 1024 coefficients using templates stacked in a tall matrix $\mathbf{P}$. The analysis is complete, meaning suitable coefficient vectors can rebuild any image. Which statement is correct?

A self-inverting transform satisfies $\mathbf{P}\mathbf{P}^T = \mathbf{I}$. An image goes through analysis $\mathbf{r} = \mathbf{P}^T\boldsymbol{\ell}$ and then synthesis with $\mathbf{Q} = \mathbf{P}$. What is the reconstructed image, and what does $\mathbf{P}^T\mathbf{P}$ equal in general?

A self-inverting transform has $\mathbf{P}\mathbf{P}^T = \mathbf{I}$. An image $\boldsymbol{\ell}$ is analyzed to coefficients $\mathbf{r} = \mathbf{P}^T\boldsymbol{\ell}$ and synthesized back with $\mathbf{Q} = \mathbf{P}$. Compute the squared reconstruction error $\|\mathbf{P}\mathbf{r} - \boldsymbol{\ell}\|^2$ and write it as a number.