去噪与异常检测：基于凸函数的变分方法
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本文档《A Variational Approach to Remove Outliers and Impulse Noise》探讨了基于变分方法处理图像噪声，特别是周期性跳动噪声（impulse noise）和异常值（outliers）的一种策略。在图像处理领域，这些问题是常见的挑战，特别是在像CCD相机这类设备中，hot pixels（热像素）和dead pixels（死像素）可能导致数据质量下降。
论文发表于2004年的《数学成像与视觉》期刊，作者Milanikova阐述了一种使用非光滑数据拟合项（如绝对值函数，即ℓ1范数）和光滑正则化项相结合的成本函数来恢复信号和图像的方法。这种组合的关键在于，通过最小化这样的成本函数，可以有效地识别并处理噪声点。非光滑数据拟合项引入了对异常值的敏感性，使得算法能够区分正常的数据点和噪声点。正常数据点被精确拟合，而异常点则由正则化项提供估计，其结果不依赖于异常值的具体数值。
这种方法的优势在于它的收敛性，这意味着算法能够在迭代过程中稳定地找到最优解。通过对这些成本函数分析，作者揭示了这种方法的自然合理性，即通过最小化过程，算法能够自动检测并剔除噪声，同时保持图像边缘的清晰度，确保修复后的图像既准确又稳定。
总结来说，这篇论文提供了一个强大的工具箱，用于图像去噪处理，特别适用于处理那些含有随机突变和异常值的图像数据。其核心思想是将数据恢复问题转化为优化问题，通过引入适当的正则化策略，实现对噪声的智能抑制和数据的有效重构，这在现代计算机视觉和信号处理中具有重要的应用价值。
Variational Approach to Remove Outliers and Impulse Noise 103
otherwise z
(k)
i
is the (unique) solution of
ψ
i
z
(k)
i
+ β D
i
Q
y
z
(k)
1
,...,z
(k)
i−1
, z
(k)
i
, z
(k−1)
i+1
,...,z
(k−1)
p
= 0,
(22)
where
z
(k)
i
> 0ifξ
(k)
i
< −ψ
i
(0
+
),
z
(k)
i
< 0ifξ
(k)
i
> −ψ
i
(0
−
);
• if q < p and i ∈{q + 1,..., p}, then z
(k)
i
is the
(unique) solution of
D
i
Q
y
z
(k)
1
,...,z
(k)
i−1
, z
(k)
i
, z
(k−1)
i+1
,...,z
(k−1)
p
= 0.
(23)
Knowing the sign of z
(k)
i
in (22) is crucial because ψ
i
is discontinuous at zero. Notice that if the minimum in
(19) occurs at a point where the function is non-smooth,
the minimizer is given exactly by (21).
Theorem 2. Let F
y
:IR
p
→ IR be of the form (7) and
suppose that H1, H2, H3 and H4 are satisfied. For
k →∞, the sequence z
(k)
defined by (19) converges to
a point
ˆ
z such that F
y
(
ˆ
z) ≤ F
y
(z), for all z ∈ IR
p
.
An important argument in the proof of this theorem
is the lemma given next.
Lemma 2. There is a constant η>0 such that for
every k ∈ IN,
F
y
z
(k)
1
,...,z
(k)
i−1
, z
(k−1)
i
, z
(k−1)
i+1
,...,z
(k−1)
p
− F
y
z
(k)
1
,...,z
(k)
i−1
, z
(k)
i
, z
(k−1)
i+1
,...,z
(k−1)
p
≥ βη
z
(k−1)
i
− z
(k)
i
2
, ∀i ∈{1,..., p}.
2.3. Total Variation Cost-Function for 1D Signals
When x is an one-dimensional signal, the discrete form
of the total-variation cost-function reads
F
y
(x) =Ax − y
2
+ β
p−1
i=1
|x
i
− x
i+1
|,
where A ∈ IR
r×p
is given. Let the entries of G ∈ IR
p×p
read G
i,i
= 1, G
i,i+1
=−1 and G
i, j
= 0if j < i or
j > i + 1, for all i =1,..., p − 1, and G
p,i
=1/ p,
for all i = 1,..., p. Using the change of variables
z = Gx,
F
y
(z) = F
y
(G
−1
z) = β
p−1
i=1
|z
i
|+Q
y
(z),
where Q
y
(z) =Bz − y
2
and B = AG
−1
.
For every i = 1,..., p, the i th column of B is denoted
b
i
and is supposed non-zero. Assumptions H1, H2, H3
and H4 clearly hold. The minimization scheme follows
from (21) and (22). For every k ∈ IN,
• if i ∈{1,..., p −1}, calculate ξ
(k)
i
=−2b
T
i
y
2b
T
i
B(z
(k)
1
, z
(k)
2
,...,z
(k)
i−1
, 0, z
(k−1)
i+1
,...,z
(k−1)
p
);
if
ξ
(k)
i
≤ β, then z
(k)
i
= 0,
if ξ
(k)
i
< −β, then z
(k)
i
=−
ξ
(k)
i
+ β
2b
i
2
> 0,
if ξ
(k)
i
>β, then z
(k)
i
=−
ξ
(k)
i
− β
2b
i
2
< 0;
• z
(k)
p
=−
ξ
(k)
p
2b
p
2
.
Notice that for images H2 is not satisfied, so this method
cannot be applied.
3. Detection and Smoothing of Outliers
3.1. Proposed Method
We will process data y ∈ IR
p
corrupted as mentioned
in I1–I2 in Section 1 by minimizing
F
y
(x) =
p
i=1
|x
i
− y
i
|+βQ(x), where
Q(x) =
1
2
p
i=1
j∈N
i
ϕ(x
i
− x
j
). (24)
Here N
i
denotes the set of the neighbors of i , for every
i = 1,...,p.Asusually [5, 15], we have i ∈ N
i
and
j ∈ N
i
⇔ i ∈ N
j
, for all i and j .Ifx is an 1D signal,
N
i
={i, i + 1},ifi = 2,...,p − 1; for a 2D image,
N
i
is the set of the 4, or the 8 pixels adjacent to i.An
illustration is given in Fig. 1(a). Put
N =
p
max
i=1
#N
i
, (25)
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