Advanced Image Denoising Methods: TV, NLM, and BM3D

Zhifei Zhang

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Outline

• TV [Rudin, et al. 1992]:

Total Variation minimization

• NLM [Buades, et al. 2005]:

Non-Local Means

• BM3D [Dabov, et al. 2006/2007]:

Block-Matching and 3D filter

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Total Variation minimization (TV)

Observed imageClean Image

Additive noise

, , ,g x y f x y n x y

f̂ g h Smooth kernel (local method)

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Total Variation minimization (TV)Global method:

=

124 100 3069 80 20066 92 211

⋯

⋮

Lower TV

2 2

,

ˆ ˆ ˆ ˆ ˆ( 1, ) ( , ) ( , 1) ( , )x y

TV f f x y f x y f x y f x y

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Total Variation minimization (TV)

Fidelity term

2

ˆ 2

1 ˆ ˆarg min2f

g f TV f

g 𝒇, 𝝀 = low 𝒇, 𝝀 = high 𝒇, 𝝀 = ∞

𝒇 =

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Total Variation minimization (TV)

𝝈 = 12

𝝈 = 25

• Straight edges are maintained. • Details and texture can be over

smoothed if 𝝀 is too large.

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Non-Local Means (NLM)

p

q1

q2

q3

,

( , ) , ,xy

i j

NLM x y w i j g i j

,

0 , 1, , 1xy xy

i j

w i j w i j

2

2

2

2

2

2

( )exp

,

( )exp

xy ij

xy

xy ij

ij

g g

w i j

g g

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Non-Local Means (NLM)

Noisy image𝝈 = 𝟐𝟎

Gaussian kernel TV NLM

Local Global Non-Local

• Preserve straight edges, as well as details and texture.

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Block-Matching and 3D filter (BM3D)

Block matching + 3D transform

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Block-Matching and 3D filter (BM3D)

Block matching + 3D transform

• Element-wise averaging• Identical blocks• Multiple blocks

• 3D transform (e.g., DWT, DFT, DCT)

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Block-Matching and 3D filter (BM3D)

• 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.

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Block-Matching and 3D filter (BM3D)

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Block-Matching and 3D filter (BM3D)

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