3D变换域稀疏协作过滤：图像去噪新方法

denoising

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本文档《Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering》主要探讨了一种创新的图像去噪方法，该方法利用了三维变换域的稀疏表示增强。传统图像处理中，图像通常被划分为二维块进行分析，但作者提出了一种将相似的二维片段组织成三维数据阵列，即“组”的策略。这种方法被称为三维变换域协作过滤（3-D Transform-Domain Collaborative Filtering）。首先，通过对图像进行三维变换，如离散余弦变换（DCT）或小波变换等，将原始图像分解为一系列系数矩阵。这些矩阵代表了不同尺度和方向上的图像特征。通过将相邻的二维块组合成三维组，可以捕捉到局部纹理和结构的共同模式，从而增加系数矩阵在三维空间中的稀疏性。这种稀疏表示有助于区分信号和噪声，因为噪声通常表现为非零系数的随机分布，而信号通常具有较少的显著项。接下来，进行一个称为收缩（shrinkage）的过程，这一步骤是通过某种阈值函数来减小或删除系数矩阵中低于某个阈值的非零元素，进一步减少噪声的影响。这个过程可以采用软阈值、硬阈值或者更复杂的统计方法，目的是尽可能地保留重要的信号成分，同时抑制噪声。最后，通过逆3-D变换将处理后的3-D系数矩阵转换回原始图像空间，得到的是联合滤波后的组内图像块。由于块之间存在重叠，这种方法能够揭示共享的细节，同时保持每个块的独特特征。这种协同过滤的优势在于它能够在去除全局噪声的同时，较好地保持图像的局部结构和细节。总结来说，这篇论文提出了一种创新的图像去噪策略，它通过3-D变换域的稀疏表示和协作过滤技术，有效地提高了去噪效果，并能在保护图像细节和保留个体特征之间取得良好的平衡。这种方法在实际应用中可能广泛用于医学成像、遥感图像处理、视频去噪等领域，对于提升图像质量具有重要意义。

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DABOV et al.: IMAGE DENOISING BY SPARSE 3-D TRANSFORM-DOMAIN COLLABORATIVE FILTERING 2083

in the group. This sparsity makes the shrinkage very effective in

attenuating the noise while preserving the features of the signal.

Let us give a simple illustration of the beneﬁt of this col-

laborative shrinkage by considering the grouped image blocks

shown in Fig. 1. Let us ﬁrst consider the case when no collab-

orative ﬁltering is performed but instead a 2-D transform is ap-

plied separately to each individual block in a given group of

fragments. Since these grouped blocks are very similar, for

any of them we should get approximately the same number, say

, of signiﬁcant transform coefﬁcients. It means that the whole

group of

fragments is represented by coefﬁcients. In con-

trast, in the case of collaborative ﬁltering, in addition to the 2-D

transform, we apply a 1-D transform across the grouped blocks

(equivalent to applying a separable 3-D transform to the whole

group). If this 1-D transform has a DC-basis element, then be-

cause of the high similarity between the blocks, there are ap-

proximately

only

signiﬁcant coefﬁcients that represent the

whole group instead of

. Hence, the grouping enhances the

sparsity, which increases with the number of grouped blocks.

As Fig. 1 demonstrates, a strong similarity between small

image blocks at different spatial locations is indeed very

common in natural images. It is a characteristic of blocks that

belong to uniform areas, edges, textures, smooth intensity gra-

dients, etc. Therefore, the existence of mutually similar blocks

can be taken as a very realistic assumption when modeling

natural images, which strongly motivates the use of grouping

and collaborative ﬁltering for an image denoising algorithm.

III. A

LGORITHM

In the proposed algorithm, the grouping is realized by

block-matching and the collaborative ﬁltering is accomplished

by shrinkage in a 3-D transform domain. The used image frag-

ments are square blocks of ﬁxed size. The general procedure

carried out in the algorithm is as follows. The input noisy image

is processed by successively extracting reference blocks from

it and for each such block:

•ﬁnd blocks that are similar to the reference one (block-

matching) and stack them together to form a 3-D array

(group);

• perform collaborative ﬁltering of the group and return the

obtained 2-D estimates of all grouped blocks to their orig-

inal locations.

After processing all reference blocks, the obtained block esti-

mates can overlap, and, thus, there are multiple estimates for

each pixel. We aggregate these estimates to form an estimate of

the whole image.

This general procedure is implemented in two different forms

to compose a two-step algorithm. This algorithm is illustrated in

Fig. 3 and proceeds as follows.

Step 1) Basic estimate.

a) Block-wise estimates. For each block in the

noisy image, do the following.

i) Grouping. Find blocks that are similar

to the currently processed one and then

This is just a qualitative statement because the actual number of signiﬁcant

coefﬁcients depends on the normalization of the transforms and on the thresh-

olds used for the 2-D and 3-D cases.

stack them together in a 3-D array

(group).

ii) Collaborative hard-thresholding. Apply

a 3-D transform to the formed group,

attenuate the noise by hard-thresholding

of the transform coefﬁcients, invert the

3-D transform to produce estimates

of all grouped blocks, and return the

estimates of the blocks to their original

positions.

b) Aggregation. Compute the basic estimate of

the true-image by weighted averaging all of

the obtained block-wise estimates that are

overlapping.

Step 2) Final estimate: Using the basic estimate, perform

improved grouping and collaborative Wiener

ﬁltering.

a) Block-wise estimates. For each block, do the

following.

i) Grouping. Use BM within the basic

estimate to ﬁnd the locations of the

blocks similar to the currently processed

one. Using these locations, form two

groups (3-D arrays), one from the noisy

image and one from the basic estimate.

ii) Collaborative Wiener ﬁltering. Apply

a 3-D transform on both groups.

Perform Wiener ﬁltering on the noisy

one using the energy spectrum of the

basic estimate as the true (pilot) energy

spectrum. Produce estimates of all

grouped blocks by applying the inverse

3-D transform on the ﬁltered coefﬁcients

and return the estimates of the blocks to

their original positions.

b) Aggregation. Compute a ﬁnal estimate of the

true-image by aggregating all of the obtained

local estimates using a weighted average.

There are two signiﬁcant motivations for the second step in the

above algorithm:

• using the basic estimate instead of the noisy image allows

to improve the grouping by block-matching;

• using the basic estimate as the pilot signal for the empirical

Wiener ﬁltering is much more effective and accurate than

the simple hard-thresholding of the 3-D spectrum of the

noisy data.

Observation Model and Notation: We consider a noisy image

of the form

where is a 2-D spatial coordinate that belongs to the image

domain

, is the true image, and is i.i.d. zero-mean

Gaussian noise with variance

, . With we

denote a block of ﬁxed size

extracted from , where is

the coordinate of the top-left corner of the block. Alternatively,

we say that

is located at in . A group of collected 2-D

blocks is denoted by a bold-face capital letter with a subscript

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