使用 proximal 算法的高效图像优化 ProxImaL

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"Proximal: 有效利用 proximity algorithms 进行图像优化" 这篇由 Felix Heide、Steven Diamond、Matthias Nießner、Jonathan Ragan-Kelley、Wolfgang Heidrich 和 Gordon Wetzstein 等人发表在 ACM Transactions on Graphics (TOG) 2016 年的文章 "ProxImaL: Efﬁcient Image Optimization using Proximal Algorithms" 展示了一种新的图像处理方法，该方法基于 proximal algorithms（亲近算法），实现了高效的图像优化。Proximal algorithms 是一种在优化问题中寻找近似最小值的数值方法，特别适用于处理非凸和非线性问题，如图像处理中的噪声去除、卷积恢复和相位检索等挑战。文章的重点在于介绍如何将亲近算法应用到图像优化的整个流程，包括但不限于以下几个关键知识点： 1. **Burst Denoising**（连拍降噪）：在连续拍摄的多张照片中，由于光照变化和相机抖动，图像可能会含有噪声。Proximal algorithm 可以有效地整合这些图像，降低噪声并保留细节，提高图像质量。 2. **Poisson Deconvolution**（泊松卷积）：泊松方程是物理光学问题中的常见模型，图像处理中常用于光度重建。使用 proximal algorithms 解决泊松方程，可以更精确地进行去模糊和恢复图像的原始细节。 3. **Phase Retrieval**（相位检索）：在光学成像中，相位信息常常丢失，仅保留幅度信息。通过 proximal algorithms，可以从幅度信息中恢复出完整的相位，从而重构出高质量的图像。 4. **Proximal Operator**（亲近算子）：是 proximal algorithms 的核心组成部分，它定义了与目标优化问题相关的特定步骤，能有效地处理各种非凸和非线性函数，使得在复杂优化问题中的迭代过程更加稳定和高效。 5. **Efficient Implementation**（高效实现）：文章还讨论了如何将这些算法高效地实现为代码，并构建数据流图（DAG），优化计算效率，使得图像处理流程能够在实际系统中实时运行。 6. **Application Diversity**（应用多样性）：ProxImaL 方法不仅限于上述提到的场景，还能广泛应用于其他图像处理任务，如超分辨率、色彩校正、图像去噪、图像融合等，展示了其在图像处理领域的广阔应用前景。 "Proximal: Efficient Image Optimation Using Proximal Algorithms" 提供了一个强大且灵活的图像优化框架，通过亲近算法解决了传统方法难以处理的图像处理难题，提高了图像处理的效率和效果。这种方法对于计算机视觉、图像处理和机器学习领域的研究者和工程师来说，具有重要的参考价值。

[Udell et al

2014]. These approaches reliably solve modest size

problems, with on the order of

10, 000

s of variables, but for image

optimization problems with millions of variables these solvers be-

come infeasible due to their memory and computational cost. There

have been several different approaches towards making an optim-

ization DSL or framework that can handle large problems such as

occur in image optimization. The approach in [Diamond and Boyd

2015] extends CVXPY to recognize and exploit fast linear transforms,

such as convolution and the discrete Fourier transform. The Epsilon

framework takes advantage of fast proximal operators for individual

functions, transforming problems so they can be efficiently solved by

a variant of ADMM [Wytock et al

2015]. The TFOCS framework

makes it easy to apply a variety of proximal and first order algorithms

to optimization problems, and accommodates fast linear transforms

[Becker et al

2011]. None of these systems can compete with ex-

isting specialized solvers for individual image processing problems,

however, and they are also limited to convex problems.

3 Representing Image Optimization Problems

We model an image optimization problem as a sum of penalties

on linear transforms K

x with x ∈ R

being the unknown:

argmin

i=1

x) with K =













, (2)

where here

K ∈ R

m×n

is one large matrix that is composed of

stacked linear operators

, . . . , K

. The linear operator

∈

×n

selects a subset of

rows of

. This subset of rows is

then the input for the penalty functions f

: R

→ R.

Image optimization problems generally contain

• variables representing the image(s) to be reconstructed,

•

a forward model of image formation in terms of linear operators,

•

a penalty based on the difference of the results of this forward

model from measured data,

• and priors and constraints on the the variables.

For example, consider a slightly more complex version of the decon-

volution problem

(1)

where the convolved image

is subsampled

by a known demosaicking pattern, which we represent with the lin-

ear operator

. We formulate our problem using a sum-of-squares

error metric, f(x) = kMDx − bk

, and the penalty function:

r(x) = µk∇xk

+ (1 − µ)k∇xk

+ I

[0,∞)

(x),

where µ ∈ [0, 1], ∇ is the gradient operator, and:

[0,∞)

(x) =

(

0, if x ≥ 0

∞, otherwise.

The penalty function encodes the priors that many gradients are

zero and the pixel values are nonnegative. Problem

(3)

shows the

full optimization problem and how we represent it in the form of

Problem (2).

opt

= argmin

kMDx − bk

+ r(x) (3)

r(x) = µk∇xk

+ (1 − µ)k∇xk

+ I

[0,∞)

(x) (4)

model:

(v) = kv − bk

, K

= MD

(v) = µkvk

, K

= ∇

(v) = (1 − µ)kvk

, K

= ∇

(v) = I

[0,∞)

(v), K

= I

(5)

Note that there are other ways to represent the problem in our stand-

ard form. For example, we could use:

(v) = kMv −bk

, K

= D.

A key insight is that the choice of representation can drastically

affect the performance of the solver algorithms. We take advantage

of this fact and provide strategies to ﬁnd an optimal reformulation.

The only assumption we make about the penalty functions

, . . . , f

is that they provide a black box for evaluating the function’s proximal

operator. The proximal operator of a function f is deﬁned as:

prox

τf

(v) = argmin



f(x) +

2τ

kx − v k



where

τ > 0

and

v ∈ R

[Parikh and Boyd 2013]. The proximal

operator optimizes over the function in isolation, but incorporates

a quadratic term that can be used to link the optimization with

a broader algorithm. Many algorithms can be carried out using

proximal operators that cannot be carried out using the traditional

approach of interacting with functions by computing their gradients

and Hessians [Parikh and Boyd 2013].

Similarly, the only assumption we make about each linear operator

is that it provides a black box for evaluating the forward operator

x → K

and the adjoint operator

z → K

. This is a useful

abstraction because many linear operators that arise in optimization

problems from image processing are fast transforms, i.e., they have

methods for evaluating the forward and adjoint operator that are

more efﬁcient than standard multiplication by the operator represen-

ted as a dense or sparse matrix. Common fast transforms in image

processing include the discrete Fourier transform (DFT), convolu-

tion, and wavelet transforms; see [Diamond and Boyd 2016a] for

many more examples.

For simplicity, we assume that all linear operators are maps from

a multidimensional real space

×···×n

to another multidimen-

sional real space

×···×m

. Complex-valued linear operators

such as the DFT are represented as real valued operators using the

standard embedding of a complex vector in

×···×n

as a real

vector in R

×···×n

We call algorithms that solve Problem

(2)

using only these black

boxes proximal, matrix-free solvers. All solver algorithms in Prox-

ImaL are proximal, matrix-free solvers. ProxImaL currently sup-

ports the Pock-Chambolle algorithm, ADMM, linearized ADMM,

and half-quadratic splitting. See the supplement for detailed deriva-

tions showing that all of these methods ﬁt into our framework from

(2)

. These algorithms can solve Problem

(2)

when the functions

, . . . , f

are convex.

Much state-of-the-art image optimization makes use of nonconvex

penalty functions; however, in applications ranging from denoising

and deconvolution to burst reconstruction and registration. Patch-

based approaches and hard thresholding in particular have been very

successful for image reconstruction problems [Krishnan and Fergus

2009; Danielyan et al. 2012; Heide et al. 2014].

Surprisingly, the same proximal, matrix-free solvers that work for

convex problems yield good results for certain problems that in-

clude nonconvex penalty functions [Danielyan et al

2012; Heide

et al

2014; Hallac et al

2015]. There is often no guarantee that

the algorithms will converge (see conditions in [Ochs et al

2014]

for exceptions). Furthermore, there is no guarantee that they ﬁnd

the optimal

, but empirically for many problems with nonconvex

penalties the algorithms do produce good results in a reasonable

number of iterations.

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