仿射非局部均值图像去噪方法
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"本文档是关于‘仿射非局部均值图像去噪’的研究论文，由Vadim Fedorov和Coloma Ballester撰写，发表在IEEE Transactions on Image Processing期刊上。该工作提出了一种非局部均值去噪方法的扩展版本，尤其针对真实场景图像中的仿射不变自相似性进行有效利用。"
在图像处理领域，噪声是常见的问题，它可能由传感器噪声、传输错误或环境因素引起。图像去噪是减少这些不期望的干扰，以恢复图像原始质量的过程。传统的去噪方法如中值滤波和均值滤波在处理某些复杂情况时可能会导致细节丢失或边缘模糊。
非局部均值（Non-Local Means）去噪是一种有效的算法，它基于图像中存在全局相似性这一假设。简单来说，非局部均值算法认为，即使在远离的位置，如果两个图像块在全局范围内具有相似性，那么它们就可能共享相同的原始信号，因此可以互相借鉴来去除噪声。
然而，对于现实世界的图像，尤其是包含复杂几何结构的图像，简单的局部相似性可能不足以捕捉到所有的重要信息。这是因为图像中的对象可能会因为透视、光照变化等因素而产生仿射变换，例如旋转、缩放或倾斜。因此，原文献中提出的仿射非局部均值图像去噪方法正是为了解决这个问题。
该方法的核心是引入了一个仿射不变的补丁相似度度量。这个度量能够在比较图像补丁时自动适应其大小和形状的变化，从而更好地识别经过仿射变换的相似区域。通过这种方式，算法能够更准确地识别和利用那些在原始非局部均值方法中可能被忽视的相似性，从而提高去噪效果。
具体实现上，作者可能采用了某种仿射变换模型来估计和补偿图像补丁之间的几何差异，然后基于这些校正后的补丁计算相似性。在实际应用中，这可能导致更好的保留图像细节和纹理，同时减少噪声的残留。
总结起来，这篇论文为图像去噪提供了一个新的视角，即考虑了仿射变换对图像自相似性的影响，通过仿射不变的非局部均值方法提升了去噪性能。这种方法对于处理具有复杂几何结构的图像特别有优势，并且有可能进一步应用于图像恢复、增强和分析等其他图像处理任务中。
1057-7149 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIP.2017.2681421, IEEE
Transactions on Image Processing
FEDOROV AND BALLESTER: AFFINE NON-LOCAL MEANS IMAGE DENOISING 3
Fig. 2. Affine covariant neighborhoods (patches) computed at corresponding
points in two images taken from different viewpoints. Despite the change in
appearance, patches capture the same visual information.
Fig. 3. An affine invariant patch comparison which is achieved by normalizing
the patches to discs and aligning them with suitable rotations.
When the tensor is affine covariant, we have that
AB
u
A
(x, r) = B
u
(Ax, r). This means that the tensors can
be used to define affine covariant regions which transform
appropriately via an affinity. We will refer to these regions as
shape-adaptive or elliptical patches to distinguish them from
the square patches of fixed size. Fig. 2 illustrates the patches
defined by the affine covariant tensors of [13], computed for
a set of corresponding points in two images related by a
homography. Note that even though the transformation is not
an affinity, the patches still match, since a homography can be
locally approximated by an affinity.
Affine covariant tensors at two corresponding locations
allow us to extract the affine distortion between corresponding
elliptical patches, up to a rotation, as shown in [13]. Indeed, for
any affine transformation A, there exists an orthogonal matrix
R such that
A = T
u
(Ax)
−
1
2
RT
u
A
(x)
1
2
. (5)
This last equation provides an intuitive geometric relationship
between the tensors, the associated elliptical patches and the
affinity. Consider a point x and the corresponding affine
covariant elliptical patch B
u
A
(x). Mapping B
u
A
(x) by the
affinity yields B
u
(Ax). The application of A can be decom-
posed in three steps. First, applying T
u
A
(x)
1
2
, we transform
B
u
A
(x) into a disc or radius r . We refer to the resulting
patch as a normalized patch. Then, a rotation is applied to
the normalized patch. Finally, T
u
(Ax)
−
1
2
maps the rotated
normalized patch to the elliptical patch B
u
(Ax).
To fully determine the affinity A, one needs to find the
rotation R. Any rotation would yield an affinity that maps the
elliptical patch associated to T
u
A
at x to the one associated
to T
u
at Ax. For a wrong value of the rotation, the image
content inside both patches will not match. Therefore, the
right value for the rotation can be computed by aligning the
image content of both patches. For this aim, we decompose the
rotation as R = R
u
(Ax)R
−1
u
A
(x), where R
u
(Ax) and R
u
A
(x)
are estimated from the image content inside the patches.
In practice, we calculate them by aligning the dominant
orientation of the normalized patches to the horizontal axis.
To compute the dominant orientation we use histograms of
gradient orientations as in the SIFT descriptors [22].
C. Computation of Affine Covariant Tensors
In this work, we use the particular dense field of affine co-
variant tensors and associated neighborhoods proposed in [13],
which are computed from the following iterative algorithm:
T
(k)
u
(x) =
R
B
(k−1)
u
(x,r)
Du(y) ⊗ Du(y) dy
Area(B
(k−1)
u
(x, r))
, (6)
where u is the given image and B
(k)
u
is the elliptical patch
related to T
(k)
u
, defined by
B
(k)
u
(x, r) = {y : hT
(k)
u
(x)(y − x), (y − x)i ≤ r
2
} (7)
for k ≥ 1, and
B
(0)
u
(x, r) = {y : |Du(x)
T
(y − x)| ≤ r} (8)
for k = 0. To simplify notation in the following, let us
denote by T
u
(x) the affine covariant structure tensor T
(k)
u
(x)
for a fixed value of k (in our experiments k = 30, as
in [13]) and a fixed value of r (notice that r > 0 is a free
parameter). Similarly, we denote by B
u
(x) the affine covariant
neighborhood B
(k)
u
(x, r).
D. An Affine Invariant Patch Similarity
We are interested in comparing patches of an image. Let us
generalize and consider the problem of comparing the patches
around two points x, y defined in images u and v, respectively.
We begin with an intuition behind the proposed similarity
measure. The patches are defined by the local metric given
by the tensors T
u
(x) and T
v
(y). In order to compare both
patches, equation (5) suggests the following mapping between
them:
C(x, y) = T
v
(y)
−
1
2
R
v
(y)R
−1
u
(x)T
u
(x)
1
2
. (9)
We can interpret C(x, y) as an affinity, mapping the elliptical
patch associated to T
u
(x) into the one associated to T
v
(y).
If u in the vicinity of x is an affinely transformed version
of v in the vicinity of y, then C(x, y) recovers the true
affinity. Intuitively, an affine invariant patch similarity could
be computed as the distance between the elliptical patch at y
and the result of applying C(x, y) to the patch at x. In practice,
it is more suitable to transform both elliptical patches to discs
of radius r (as depicted in Fig. 3) and to compare the aligned
normalized patches:
D
a
(t, x, y) =
Z
∆
t
g
t
(h)
u(x + T
−
1
2
u
R
u
(x)h) − v(y + T
−
1
2
v
R
v
(y)h)
2
dh,
(10)
where ∆
t
denotes a disc centered at the origin with radius
proportional to t > 0 (the so-called scale) and big enough
such that the weighting function g
t
has effective support in
∆
t
. D
a
is a multiscale affine invariant patch distance which
intrinsically extends the set of available patches.
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