Modular ADMM-Based Strategies for Optimized Compression, Restoration, and Distributed Representations of Visual Data

Abstract

Iterative techniques are a well-established tool in modern imaging sciences, allowing to address complex optimization problems via sequences of simpler computational processes. This approach has been significantly expanded in recent years by iterative designs where explicit solutions of optimization subproblems were replaced by black-box applications of ready-to-use modules for denoising or compression. These modular designs are conceptually simple, yet often achieve impressive results. In this chapter, we overview the concept of modular optimization for imaging problems by focusing on structures induced by the alternating direction method of multipliers (ADMM) technique and their applications to intricate restoration and compression problems. We start by emphasizing general guidelines independent of the module type used and only then derive ADMM-based structures relying on denoising and compression methods. The wide perspective on the topic should motivate extensions of the types of problems addressed and the kinds of black boxes utilized by the modular optimization. As an example for a promising research avenue, we present our recent framework employing black-box modules for distributed representations of visual data.

Appendix: Operational Rate-Distortion Optimizations in Block-Based Architectures

The computational challenge of operational rate-distortion optimizations (see section “Preliminaries: Lossy Compression via Operational Rate-Distortion Optimization”) is often addressed via the squared-error metric

$$\displaystyle \begin{aligned} D\left( \mathbf{x}, \mathbf{v} \right) = \left\|{ \mathbf{x} - \mathbf{v} } \right\|{}_2^2,\end{aligned} $$

(47)

leading to practical forms of the Lagrangian rate-distortion optimization (25). These useful structures also process the signal x based on its segmentation into a set of nonoverlap** blocks $\left \lbrace {\mathbf {x}}_i \right \rbrace _{i\in \mathcal {I}}$; here, each of them is a column vector of N_b samples, and $\mathcal {I}$ is the set of indices corresponding to the nonoverlap** blocks of the signal. Correspondingly, the decompressed signal v is decomposed into a set of nonoverlap** blocks $\left \lbrace {\mathbf {v}}_i \right \rbrace _{i\in \mathcal {I}}$. This lets us casting (47) into

$$\displaystyle \begin{aligned} D\left( \mathbf{x}, \mathbf{v} \right) = \sum_{ i\in\mathcal{I} } \left\|{ {\mathbf{x}}_i - {\mathbf{v}}_i } \right\|{}_2^2 , \end{aligned} $$

(48)

exhibiting that, for squared-error measures, the total distortion can be computed as the sum of distortions associated with its nonoverlap** blocks. While this property is satisfied for any segmentation of the signal into nonoverlap** blocks, we will exemplify it here for blocks of equal sizes that allow using one block-level compression procedure for all the blocks.

Mirroring the definitions described in section “Preliminaries: Lossy Compression via Operational Rate-Distortion Optimization” for full-signal compression architectures, the block-level process corresponds to a function $C_b: \mathbb {R}^{N_b} \rightarrow \mathcal {B}_b$, map** the N_b-dimensional signal-block domain to a discrete set $\mathcal {B}_b$ of binary compressed representations of blocks. The associated block decompression process is denoted by the function $F_b: \mathcal {B}_b \rightarrow \mathcal {S}_b$, map** the binary compressed representations in $\mathcal {B}_b$ to their decompressed signal blocks from the discrete set $\mathcal {S}_b \subset \mathbb {R}^{N_b}$. The bit-cost evaluation function $R_b\left ( {\mathbf {v}}_i \right )$ is defined to quantify the number of bits needed for the compressed representation matching the decompressed signal block ${\mathbf {v}}_i \in \mathbb {R}^{N_b}$. Then, the compression of the nonoverlap** signal blocks $\left \lbrace {\mathbf {x}}_i \right \rbrace _{i\in \mathcal {I}}$ producing the decompressed blocks $\left \lbrace {\mathbf {v}}_i \right \rbrace _{i\in \mathcal {I}}$ requires a total bit budget satisfying

$$\displaystyle \begin{aligned} R\left( \mathbf{v} \right) = \sum_{ i\in\mathcal{I} } R_b\left( {\mathbf{v}}_i \right). \end{aligned} $$

(49)

Plugging the block-based compression design into the Lagrangian form (25) gives

(50)

that reduces to a set of block-level rate-distortion Lagrangian optimizations, i.e.,

(51)

Note that the block-level optimizations in (51) are independent and refer to the same Lagrangian multiplier λ. Commonly, compression designs are based on processing of low-dimensional blocks, allowing to practically address the block-level optimizations in (51). For example, one can evaluate the Lagrangian cost for all the elements in $\mathcal {S}_b$ (since this set is sufficiently small).