二维轮廓波变换：捕捉图像边缘几何的理想工具

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标题：“轮廓线变换（Contourlet Transform）：一种高效的方向性多分辨率图像表示” 描述：在传统的图像处理中，如傅立叶变换和小波变换，由于它们的一维扩展性质，往往在捕捉图像边缘的几何结构方面存在局限性。为了更有效地提取视觉信息中的关键内在几何结构，该论文提出了一种名为“轮廓线变换”的真正二维变换。与曲线let等方法先在连续域开发变换然后对采样数据进行离散化不同，该研究采用了离散域构建，然后研究其向连续域扩张的收敛性。具体来说，通过非分离滤波器银行构建了一个多分辨率和多方向的离散域扩展，这种方法类似于小波由滤波器银行导出的过程。这种构造产生了基于轮廓段的灵活多分辨率、局部和方向性的图像展开，这使得轮廓线变换能够更好地适应图像中的复杂几何特性。算法上，轮廓线变换具有快速的迭代滤波器银行算法，对于N像素的图像，操作次数为O(N)，这在计算效率上具有显著优势。更重要的是，论文建立了滤波器银行与连续域轮廓线变换之间的精确联系，通过一个方向性多分辨率分析框架，确保了其理论上的完备性。作者证明了，当采用抛物线缩放和足够的方向性消失矩时，轮廓线变换能够达到最优的逼近率，适用于具有沿着可微曲线上断点的分段光滑函数。这意味着它在处理含有边缘和角点的图像时，能够提供更精确的重构效果。最后，通过一系列数值实验，作者展示了轮廓线变换在图像处理应用中的潜力，包括边缘检测、图像压缩、图像分割以及可能的其他信号处理任务，证明了其在实际问题中的有效性。轮廓线变换作为一种新型的图像表示工具，结合了多分辨率和方向性分析的优势，为处理复杂几何结构提供了强大的支持。

4 IEEE TRANSACTIONS ON IMAGE PROCESSING

M M

(a)

M M

ˆx

(b)

Fig. 2. Laplacian pyramid. (a) One level of decomposition. The outputs are a coarse approximation a[n] and a difference b[n] between the original signal

and the prediction. (b) The new reconstruction scheme for the Laplacian pyramid [26].

to its translates with respect to the sampling lattice by M )

provides a tight frame with frame bounds are equal to 1. In this

case, we proposed the use of the optimal linear reconstruction

using the dual frame operator (or pseudo-inverse) as shown

in Figure 2(b). The new reconstruction differs from the usual

method, where the signal is obtained by simply adding back

the difference to the prediction from the coarse signal, and

was shown [26] to achieve signiﬁcant improvement over the

usual reconstruction in the presence of noise.

C. Iterated directional ﬁlter banks

Bamberger and Smith [24] constructed a 2-D directional

ﬁlter bank (DFB) that can be maximally decimated while

achieving perfect reconstruction. The DFB is efﬁciently im-

plemented via an l-level binary tree decomposition that leads

to 2

subbands with wedge-shaped frequency partitioning as

shown in Figure 3(a). The original construction of the DFB in

[24] involves modulating the input image and using quincunx

ﬁlter banks with diamond-shaped ﬁlters [27]. To obtain the

desired frequency partition, a complicated tree expanding rule

has to be followed for ﬁner directional subbands (e.g., see [28]

for details).

In [29], we proposed a new construction for the DFB that

avoids modulating the input image and has a simpler rule

for expanding the decomposition tree. Our simpliﬁed DFB

is intuitively constructed from two building blocks. The ﬁrst

building block is a two-channel quincunx ﬁlter bank [27] with

fan ﬁlters (see Figure 4) that divides a 2-D spectrum into two

directions: horizontal and vertical. The second building block

of the DFB is a shearing operator, which amounts to just

reordering of image samples. Figure 5 shows an application

of a shearing operator where a −45

◦

direction edge becomes

a vertical edge. By adding a pair of shearing operator and its

inverse (“unshearing”) to before and after, respectively, a two-

channel ﬁlter bank in Figure 4, we obtain a different directional

frequency partition while maintaining perfect reconstruction.

Thus, the key in the DFB is to use an appropriate combination

of shearing operators together with two-direction partition of

quincunx ﬁlter banks at each node in a binary tree-structured

ﬁlter bank, to obtain the desired 2-D spectrum division as

shown in Figure 3(a). For details, see [29] (Chapter 3).

Using multirate identities [8], it is instructive to view an l-

level tree-structured DFB equivalently as a 2

parallel channel

ﬁlter bank with equivalent ﬁlters and overall sampling matrices

as shown in Figure 3(b). Denote these equivalent (directional)

synthesis ﬁlters as D

(l)

, 0 ≤ k < 2

, which correspond to the

subbands indexed as in Figure 3(a). The corresponding overall

ˆx

Fig. 4. Two-dimensional spectrum partition using quincunx ﬁlter banks with

fan ﬁlters. The black regions represent the ideal frequency supports of each

ﬁlter. Q is a quincunx sampling matrix.

(a) (b)

Fig. 5. Example of shearing operation that is used like a rotation operation

for DFB decomposition. (a) The “cameraman” image. (b) The “cameraman”

image after a shearing operation.

sampling matrices were shown [29] to have the following

diagonal forms

(l)

(

diag(2

l−1

, 2) for 0 ≤ k < 2

l−1

diag(2, 2

l−1

) for 2

l−1

≤ k < 2

(3)

which means sampling is separable. The two sets correspond

to the mostly horizontal and mostly vertical set of directions,

respectively.

From the equivalent parallel view of the DFB, we see that

the family

(l)

[n −S

(l)

0≤k<2

, m∈Z

, (4)

obtained by translating the impulse responses of the equivalent

synthesis ﬁlters D

(l)

over the sampling lattices by S

(l)

provides a basis for discrete signals in l

). This basis

exhibits both directional and localization properties. Figure 6

demonstrates this fact by showing the impulse responses of

equivalent ﬁlters from an example DFB. These basis functions

have quasi-linear supports in space and span all directions. In

other words, the basis (4) resembles a local Radon transform

and are called Radonlets. Furthermore, it can be shown [29]

that if the building block ﬁlter bank in Figure 4 uses orthogonal

ﬁlters, then the resulting DFB is orthogonal and (4) becomes

an orthogonal basis.

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