精确实现的Curvelet变换：图像去噪新方法

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本文档深入介绍了Curvelet变换，一种在图像处理领域具有重要应用的数学工具，由Jean-Luc Starck, Emmanuel J. Candès, 和 David L. Donoho三位作者于2002年6月在IEEE Transactions on Image Processing发表。Curvelet变换相比于传统的图像处理方法，如 Ridgelet变换，具有更高的精确度、稳定性、易于实现以及较低的计算复杂性。文章的核心内容围绕Curvelet变换的数字实现展开。Curvelet是一种特殊的图像分解方法，它构建了一种局部化的频域分析框架，允许对图像中的细节进行精细捕捉，特别适用于处理高频信息和边缘检测。与传统的基于像素的表示不同，Curvelet通过在频率域内近似数字Radon变换，实现了对图像的多尺度分析，其中每个子带（subband）对应于不同方向和尺度的信息。关键的技术之一是傅里叶空间内的简单插值，这种插值将离散的笛卡尔坐标样本转换为极坐标网格上的样本，这个网格是基于同心圆的几何结构。尽管插值方法相对粗糙，但实际的视觉效果却出人意料地好。这表明其在实际图像处理中的性能表现优秀。在Curvelet变换的具体实现中，首先利用Ridgelet变换作为基础组件，然后构建了一个滤波器银行系统，用于生成一系列曲线小波（curvelets），这些小波在频率域具有紧凑的支持特性，从而提供了更丰富的局部特征表示。这种方法对于图像去噪、图像压缩、图像恢复等任务具有显著优势，因为它能够在保持细节的同时减少冗余信息。这篇文档为读者揭示了Curvelet变换的理论背景、数字实现方法以及其在图像处理中的优越性能，对于希望深入了解该领域的研究者和技术人员来说是一份极其有价值的学习资料。然而，由于版权问题，可能需要在图书馆获取或通过合法途径下载阅读。

672 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 11, NO. 6, JUNE 2002

. In contrast, in analyzing objects with edges, wavelets

give an

-term squared approximation error only of size ,

and wavelet thresholding gives a corresponding MSE only of

size

and no better.

D. This Paper

So according to theory for a certain continuous-space model,

discrete ridgelet transforms and discrete curvelet transforms

provide near-ideal sparsity of representation of both smooth

objects and of objects with edges. In a certain continuous-space

statistical theory, this implies that simple thresholding of noisy

coefficients in these expansions is a near-optimal method of

denoising in the presence of white Gaussian noise.

In this paper we provide an initial test of these ideas in a dig-

ital image processing setting, where images are available on an

-by- grid. We first review some basic ideas about ridgelet and

curvelet representations in the continuum. We next use these

to develop a series of digital ridgelet and digital curvelet trans-

forms taking digital input data on a Cartesian grid. Next we con-

sider a model denoising problem where we embed some stan-

dard images in white noise and apply thresholding in the digital

curvelet transform domain. Finally we discuss interpretations

and possibilities for future work.

Not surprisingly, other researchers have undertaken efforts to

implement ridgelet and curvelet transforms, and develop appli-

cations. In addition to work mentioned in the body of the article,

we would like to point out the work of Do and Vetterli [13],

Donoho and Duncan [18]. We would also like to mention the

articles of Sahiner and Yagle [21]–[23], Olson and DeStefano

[20], Zhao et al. [30], and Zuidwijk [31], [32] although these

references are less directly related.

II. C

ONTINUOUS RIDGELET TRANSFORM

The 2-D continuous ridgelet transform in can be defined

as follows [2]. We pick a smooth univariate function

with sufficient decay and satisfying the admissibility condition

(1)

which holds if, say,

has a vanishing mean .We

will suppose that

is normalized so that .

For each

, each and each , we define

the bivariate ridgelet

(2)

this function is constant along lines

const

Transverse to these ridges it is a wavelet. Given an integrable

bivariate function

, we define its ridgelet coefficients by

We have the exact reconstruction formula

(3)

valid a.e. for functions which are both integrable and square in-

tegrable. Furthermore, this formula is stable as one has a Par-

seval relation

(4)

Hence, much like the wavelet or Fourier transforms, the identity

(3) expresses the fact that one can represent any arbitrary func-

tion as a continuous superposition of ridgelets. Discrete analogs

of (3) and (4) exist; see [2] or [17] for a slightly different ap-

proach.

A. Radon Transform

A basic tool for calculating ridgelet coefficients is to view

ridgelet analysis as a form of wavelet analysis in the Radon do-

main. We recall that the Radon transform of an object

is the

collection of line integrals indexed by

given by

(5)

where

is the Dirac distribution. The ridgelet coefficients

of an object are given by analysis of the Radon

transform via

Hence, the ridgelet transform is precisely the application of a

one-dimensional (1-D) wavelet transform to the slices of the

Radon transform where the angular variable

is constant and

is varying.

B. Ridgelet Pyramids

Let

denote a dyadic square

and let be the collection of all such

dyadic squares. We write

for the collection of all dyadic

squares of scale

. Associated to the squares we con-

struct a partition of energy as follows. With

a nice smooth

window obeying

, we dilate

and transport

to all squares at scale , producing a collec-

tion of windows

such that the s, , make up a

partition of unity. We also let

denote the transport operator

acting on functions

via

With these notations, it is not hard to see that

and, therefore, summing the above equality across squares at a

given scale gives

(6)

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