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这篇文章的标题是《一种凸多变量噪声模型的非线性反尺度空间方法》，描述指出这是一篇关于图像复原算法的研究，特别是基于TV（Total Variation，全变分）正则化的去噪模型，旨在有效去除图像噪声。从这些信息中，我们可以提炼出以下几个关键知识点： 1. 图像复原与去噪图像复原是图像处理中的一个重要领域，目的是尽可能恢复出质量更高、噪声更少的图像。去噪是图像复原的一个子领域，专注于从图像中移除或减少噪声成分，以提高图像质量。图像噪声通常是由成像设备、传输过程中的干扰或环境因素造成的。 2. TV正则化模型 TV正则化是一种广泛应用于图像处理中的方法，尤其是在图像去噪领域。它的核心思想是将图像看作是二维或三维信号的变分问题，并通过最小化图像的全变分来求解，其中全变分度量了图像像素或体素值的变化。TV正则化能够保留图像的边缘信息，因此在去除噪声的同时尽量减少边缘模糊。 3. 凸多变量噪声模型凸多变量噪声模型是指在噪声模型中，噪声项具有凸性质，这使得优化问题的解具有较好的全局唯一性。文章中提到的多变量噪声模型和早期的加性噪声模型不同，其特点是噪声是乘性的，与图像信号相乘，而非简单相加。这种模型更符合一些实际图像信号处理场景，比如在遥感图像处理中，噪声往往与图像信号的亮度成正比。 4. 非线性反尺度空间方法尺度空间理论是图像处理中的一种理论框架，用于表征图像的多尺度特性。非线性反尺度空间方法是一种用于图像处理的数学技术，它基于尺度空间理论，通过逐步去除图像信号的细节特征来达到去噪的目的。文章中提出的非线性反尺度空间方法能够有效地处理凸多变量噪声模型，并展示了其在去噪上的优越性能。 5. Bregman距离 Bregman距离是一种用于优化问题的数学度量，用于衡量解的迭代过程中目标函数值的变化。在本文中，研究者展示了所提出的非线性反尺度空间方法与Bregman距离的关系，这有助于理解算法在去除噪声的同时如何维护图像的全局结构。 6. 收敛性与正则化效应文章中证明了提出的方法对于多变量噪声模型的收敛性，并讨论了其正则化效应。这意味着算法不仅能够有效去除噪声，还能保证图像的质量不会因去噪过程而过度退化。从以上知识点可以看出，这篇文章是在图像处理领域的深入研究，它不仅回顾了现有的一些方法，并提出了一种新的针对多变量噪声模型的TV正则化框架。通过数值实验验证了新方法在去除噪声方面的优异性能和对早期多变量噪声模型的显著改进。此外，研究者还探讨了算法的参数依赖性，为实际应用提供了理论依据和指导。

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SIAM J. IMAGING SCIENCES



2008 Society for Industrial and Applied Mathematics

Vol. 1, No. 3, pp. 294–321

A Nonlinear Inverse Scale Space Method

for a Convex Multiplicative Noise Model

∗

Jianing Shi

†

and Stanley Osher

‡

Abstract. We are motivated by a recently developed nonlinear inverse scale space method for image denoising

[M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179–212; M. Burger,

S. Osher, J. Xu, and G. Gilboa, in Variational, Geometric, and Level Set Methods in Computer Vi-

sion, Lecture Notes in Comput. Sci. 3752, Springer, Berlin, 2005, pp. 25–36], whereby noise can be

removed with minimal degradation. The additive noise model has been studied extensively, using the

Rudin–Osher–Fatemi model [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259–268],

an iterative regularization method [S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, Multi-

scale Model. Simul., 4 (2005), pp. 460–489], and the inverse scale space ﬂow [M. Burger, G. Gilboa,

S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179–212; M. Burger, S. Osher, J. Xu, and

G. Gilboa, in Variational, Geometric, and Level Set Methods in Computer Vision, Lecture Notes

in Comput. Sci. 3752, Springer, Berlin, 2005, pp. 25–36]. However, the multiplicative noise model

has not yet been studied thoroughly. Earlier total variation models for the multiplicative noise

cannot easily be extended to the inverse scale space, due to the lack of global convexity. In this

paper, we review existing multiplicative models and present a new total variation framework for the

multiplicative noise model, which is globally strictly convex. We extend this convex model to the

nonlinear inverse scale space ﬂow and its corresponding relaxed inverse scale space ﬂow. We demon-

strate the convergence of the ﬂow for the multiplicative noise model, as well as its regularization

eﬀect and its relation to the Bregman distance. We investigate the properties of the ﬂow and study

the dependence on ﬂow parameters. The numerical results show an excellent denoising eﬀect and

signiﬁcant improvement over earlier multiplicative models.

Key words. inverse scale space, total variation, multiplicative noise, denoising, Bregman distance

AMS subject classiﬁcations. 35-xx, 65-xx

DOI. 10.1137/070689954

1. Introduction. Image denoising is an important problem of interest to the mathemat-

ical community that has wide application in ﬁelds ranging from computer vision to medical

imaging. A variety of methods have been proposed over the last decades, including traditional

ﬁltering, wavelets, stochastic approaches, and PDE-based variational methods. We refer the

reader to [9] for a review of various methods. The additive noise model has been extensively

studied, using the Rudin–Osher–Fatemi (ROF) model [23], an iterative regularization method

[21], and the inverse scale space (ISS) ﬂow [5, 6]. However, the multiplicative noise has not yet

been studied thoroughly. In this paper, we obtain a new convex multiplicative noise model

and extend it to the nonlinear ISS.

∗

Received by the editors April 30, 2007; accepted for publication (in revised form) May 30, 2008; published

electronically September 4, 2008. This research was supported by NSF grants DMS-0312222 and ACI-0321917 and

by NIH G54 RR021813.

http://www.siam.org/journals/siims/1-3/68995.html

†

Department of Biomedical Engineering, Columbia University, 351 Engineering Terrace, MC 8904, 530 W. 120th

St., New York, NY 10027 (js2615@columbia.edu).

‡

Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095 (sjo@math.ucla.edu).

294

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INVERSE SCALE SPACE FOR MULTIPLICATIVE NOISE 295

In the additive noise model, one sought to recover the signal u which was corrupted by

additive noise v; i.e., f = u+v. The ROF model [23] introduced total variational minimization

to image processing and deﬁned the solution as follows:

(1.1) u = arg min

u∈BV (Ω)



|u|

f −u



for some scale parameter λ>0, where BV (Ω) denotes the space of functions with bounded

variation on Ω, equipped with the BV seminorm which is formally given by |u|



|∇u|,

also referred to as the total variation (TV) of u. Successful implementations of this mini-

mization problem include the Euler–Lagrange equation by gradient descent on the original

minimization problem [23], second-order cone programming [15], duality [8], and an extremely

fast method based on graph cuts [11]. The choice of the positive scale constant λ in the ROF

model is important in that large λ corresponds to a small amount of noise removal, while

small λ can result in oversmoothing the image. The ROF model was extended to an iterative

regularization method based on the Bregman distance in [21], motivated by Meyer’s analysis

in [19], where he deﬁned texture, “highly oscillatory patterns in an image,” as elements of

the dual space of BV (Ω). In order to preserve the texture information Meyer suggested a

modiﬁed variational problem using the space (G,·

∗

), the dual space for the closure of the

Schwartz space in BV (Ω), equipped with the 

∗

(see [19]), as

(1.2) u = arg min

u∈BV (Ω)

{|u|

+λf −u

∗

Meyer also used his analysis to quantify and explain the observed loss of contrast in the ROF

model.

The iterative regularization model, introduced in [21], improved the quality of regularized

solutions in terms of texture preservation as well as the signal-to-noise ratio (SNR). The

nonlinear ISS method devised in [5, 6] formulates the iterative regularization method as a

continuous time ﬂow and provides a better temporal resolution and a better stopping criterion

and is usually much faster to implement than the iterative regularization. Both methods have

solutions which start from an initial condition u

(x) = 0 (assuming



f = 0), come close to the

true u, and then approach the noisy image f. The idea behind these two methods is that larger

scales converge faster than smaller ones, where scale can be precisely deﬁned using Meyer’s



∗

. The ISS method has proved to yield better denoising results than standard ROF. In fact

it yields improved denoising results among PDE-based methods [4, 5, 6, 9, 21, 23].

Multiplicative noise has not been as well studied. We consider the problem of seeking the

true signal u from a noisy signal f, corrupted by multiplicative noise η. This arises in medical

imaging, e.g., magnetic ﬁeld inhomogeneity in MRI [14, 20], speckle noise in ultrasound [28],

and speckle noise in synthetic aperture radar (SAR) images [2, 27]. Denoising of speckled

ultrasound images has been tackled, e.g., in [1], and more general noise models, which are

correlated with the signal amplitude (such as multiplicative noise), have been studied in

[24]. The ﬁrst total variation approach to solving the multiplicative model was presented in

[22], which used a constrained optimization approach with two Lagrange multipliers. The

authors also considered blurry and noisy images. However, their ﬁtting term is nonconvex,

in general, which leads to diﬃculties in using the iterative regularization or the ISS method.

Downloaded 12/01/12 to 125.71.231.223. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

296 JIANING SHI AND STANLEY OSHER

In [2] a multiplicative model was introduced involving BV , with a ﬁtting term derived from

a maximum a posteriori (MAP) estimation. However, the ﬁtting team was convex only for

u ∈ (0,2f). Some interesting analysis for the convergence, existence, and uniqueness of the

solution was done, though the numerical results exhibited some loss of contrast.

Motivated by the eﬀectiveness of the ISS, we derive a TV framework that is globally

convex. We further construct a new ISS method for the multiplicative noise based on this

convex functional.

2. Review of multiplicative models. We are motivated by the following problem in image

restoration. In what follows we will always require f>0,u>0. Given a noisy image f :Ω→ R,

where Ω is a bounded open subset of R

, we want to obtain a decomposition

(2.1) f = uη ,

where u is the true signal and η the noise. We would like to denoise the signal while preserving

the maximum information about u. In image processing, such information is manifested in a

large part by keeping the sharpness of edges, since conventional denoising methods tend to

blur the image by smearing the edge information or by creating ringing artifacts at edges.

We assume that we have some prior information about the mean and variance of the

multiplicative noise,



η =1,



(η −1)

= σ

,(2.2)

where N =



1. This states that the mean of the noise is equal to 1, and the variance is equal

to σ

The procedure introduced in [22] sought the solution of a constrained optimization problem

in the space of bounded variation, implemented by the gradient projection method involving

Euler–Lagrange equations and artiﬁcial time evolution as in [23]. The following optimization

problem was studied:

(2.3) u = arg min

u∈BV (Ω)

{J(u)+λH(u,f)},

where J(u)=



|∇u| and H(u,f) is a ﬁdelity term for a known signal or image f. In the case

of additive noise, H(u,f)=

f −u

, whereas H(u,f) takes a more sophisticated form for

multiplicative noise. A ﬁdelity term H(u,f) was used, consisting of two integrals with two

Lagrange multipliers H(u,f)=λ



+λ



(

−1)

. The initial data was chosen to satisfy the

constraints



u(0)

=1,





u(0)

−1



= σ

,(2.4)

where N =



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INVERSE SCALE SPACE FOR MULTIPLICATIVE NOISE 297

A gradient projection method was introduced to make sure the two constraints are always

satisﬁed during the evolution, which reduces to requiring

∂

∂t



= −



=0(2.5)

and

∂

∂t





−1



=0=

∂

∂t







= −2



.(2.6)

Thus we evolve the following Euler–Lagrange equation:

(2.7) u

= ∇·

∇u

|∇u|

+λ

where the values of λ

and λ

can be found dynamically using the gradient projection method.

However, for this to be a convex model we need λ

,λ

≥ 0, which is not necessarily the case

as t evolves. Moreover, if we ﬁx λ

,λ

> 0, then the corresponding minimization problem will

lead us to a sequence of constant functions u approaching +∞.

Multiplicative noise, in various imaging modalities, is often not necessarily Gaussian noise.

In the speckle noise model, such as SAR imagery, the noise is treated as gamma noise with

mean equal to 1. The distribution of the noise η takes the form of g(η),

(2.8) g(η)=

Γ(L)

L−1

exp(−Lη)1

{η>=0}

Based on this, Aubert and Aujol [2] formulated the following minimization problem:

(2.9) u = arg min

u∈S(Ω)



J(u)+λ





logu +



where J(u)=



|∇u|, and new ﬁtting function H(u,f)=



(logu +

) is strictly convex for

u ∈ (0,2f). The derivation of this functional is based on MAP on p(u|f), assuming the noise η

follows a gamma law with mean 1, and p(u) follows a Gibbs prior. Some interesting analysis

of this model was done in [2].

For denoising, the ISS method has so far been applied only to the additive noise model.

In this paper we will present a new TV functional for the multiplicative noise model, which

is globally convex. We will also extend the TV minimization approach to the ISS for our new

model.

3. Inverse scale space. In this section, we review some key results for ISS. An iterative

algorithm based on TV was introduced in [21], which was preceded by several interesting and

related pieces of work [16, 25, 26]. The ISS is based on the iterative algorithm, by taking the

time step to the limit of 0, formally introduced in [5, 6] for dealing with additive noise and

deconvolution. We review some key results for the ISS in this section.

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298 JIANING SHI AND STANLEY OSHER

Following the treatment of Burger et al. [5, 6], the TV-based minimization (2.3) can

be extended into an iterative algorithm. Introduced in [21], the authors propose solving a

sequence of u

(3.1) u

= arg min

u∈BV (Ω)

{D(u,u

k−1

)+λH(u,f)},

where u

is the primal variable and D(·,·) is the Bregman distance related to a diﬀerentiable

functional J : R

→ R, formally deﬁned by

(3.2) D

(u,v)=J(u) − J(v)−u− v,p,p∈ ∂J (v ).

The initial condition u

is a constant satisfying



∂

H(u

,f)=0andp

= 0. Here ∂

H(u,f)

is the subgradient of H(u,f) with respect to u, and p ∈ ∂J (u) is an element in the subgradient

of J at u. We will also use the notation p

, which is an element of the subgradient of J at u

For a continuously diﬀerentiable functional there exist a unique element p in the sub-

diﬀerential and, consequently, a unique Bregman distance. For a nonsmooth and not strictly

convex functional such as TV, one can still obtain convergence of the reconstruction, as long

as the functional has suitable lower semicontinuity and compactness properties in the weak-*

topology of BV (Ω) (and also in L

(Ω) by compact embedding).

One can rewrite (3.1) and arrive at

(3.3) u

= arg min

u∈BV (Ω)

{J(u)−u,p

k−1

+λH(u,f)}.

The Euler–Lagrange equation for (3.3)is

−p

k−1

+λ∂

H(u

,f)=0.

Such an iterative regularization is taken to the limit by letting λ =Δt, u

= u(kΔt), and then

dividing by Δt and letting Δt → 0. This leads to a continuous ﬂow in the ISS,

(3.4) ∂

p = −∂

H(u,f),p∈ ∂J (u),

with the initial conditions p

=0, u

= c

, where c

is a constant satisfying



∂

H(c

,f)=0.

This was introduced in [21] and analyzed in [4].

This elegant formulation of the ISS ﬂow is not straightforward to compute in two or more

dimensions, since the relation between p and u is complicated for TV and other nonlinear

cases. However, in one dimension one can solve this directly by regularizing the TV term

J(u) by the following:

(3.5) J



(u)=





|∇u|

+,

where >0 is a small constant.

A relaxed form of the ISS ﬂow was introduced, which involves the following coupled

equations:

(3.6)

∂

u = −p+λ(−∂

H(u,f)+v),

∂

v = −α∂

H(u,f),

with p ∈ ∂J (u) and initial conditions u

= c

=0.

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