Autocorrelation-Driven Diffusion Filtering Linköping University Post Print

Autocorrelation-Driven Diffusion Filtering Linköping University Post Print
Autocorrelation-Driven Diffusion Filtering
Michael Felsberg
Linköping University Post Print
N.B.: When citing this work, cite the original article.
©2011 IEEE. Personal use of this material is permitted. However, permission to
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Michael Felsberg , Autocorrelation-Driven Diffusion Filtering, 2011, IEEE Transactions on
Image Processing, (20), 7, 1797-1806.
http://dx.doi.org/10.1109/TIP.2011.2107330
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-65430
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1
Autocorrelation-Driven Diffusion Filtering
Michael Felsberg, Member, IEEE
Abstract—In this paper we present a novel scheme for
anisotropic diffusion driven by the image autocorrelation function. We show the equivalence of this scheme to a special case of
iterated adaptive filtering. By determining the diffusion tensor
field from an autocorrelation estimate, we obtain an evolution
equation that is computed from a scalar product of diffusion
tensor and the image Hessian. We propose further a set of filters
to approximate the Hessian on a minimized spatial support. On
standard benchmarks, the resulting method performs favorable
in many cases, in particular at low noise levels. In a GPU
implementation video real-time performance is easily achieved.
Index Terms—Diffusion filtering, adaptive filtering, structure
tensor, steerable filters, image enhancement
EDICS: TEC-PRC, TEC-PDE, TEC-RST
I. I NTRODUCTION
T
HE efficient structure-preserving image denoising and
enhancement is a relevant problem in many applications.
In this paper, we propose a novel variant of tensor-driven
diffusion filtering that: (a) only requires a scalar product of
diffusion tensor and Hessian in the evolution equation, (b) is
equivalent to a particular instance of adaptive filtering, and (c)
is easy to implement in a fast numerical scheme.
Structure-preserving denoising and enhancement are practical problems to be solved in many applications, e.g. image and
video coding, digital cameras, and medical imaging. Most of
these applications require a real-time approach, which limits
the range of available methods. The development of real-time
capable approaches with state-of-the-art performance is still a
relevant topic. The theoretical and practical insights formulated
in this paper allow to achieve this goal, as it is shown in the
experiments.
Structure-preserving denoising has a rather long history
and non-linear diffusion [1] is probably one of the most
commonly used techniques. The anisotropic extension of nonlinear diffusion using the structure tensor [2] often leads to
better results, in particular close to lines and edges. The
numerical implementation of anisotropic diffusion is however
less trivial as expected, as can be seen by the variety of
algorithms and publications on the topic [3], [4], [5]. This is
even more severe as we observed that the applied numerical
scheme has larger influence on the quality of results than
the choice of the method, i.e., using a sub-optimal numerical
scheme for anisotropic diffusion results in worse peak-signal
to noise ratios (PSNR) than a closer to optimal scheme for
non-linear diffusion.
Copyright (c) 2010 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained
from the IEEE by sending a request to [email protected]
M. Felsberg is with the Computer Vision Laboratory, Department of
Electrical Engineering, Linköping University, S-58183 Linköping, Sweden,
[email protected]
Besides non-linear diffusion, a large variety of approaches
for image denoising have been published, too many to be
covered here. Some of the more popular approaches use
iterated adaptive filters [6], bilateral filtering [7], [8], meanshift filtering [9], [10], channel decomposition [11], multiband filtering [12], or block-based methods [13], [14]. Optimal
results usually require estimates of image priors [15], [16],
[17]. Some of the mentioned methods have been analyzed and
compared in [15], [18], [19].
Besides the above cited work on diffusion, approaches
to trace-based diffusion filtering have been suggested [20]
and their relation to variational formulations and divergence
expressions have been investigated [21]. The main difference
of our approach to this previous work lies in the diffusion function and the spatial averaging in its computation. Tschumperlé
and Deriche compute the diffusion tensor by averaging over
different channels, i.e., in the case of grey scale images, no
averaging is done at all. In that case, the diffusion tensor
is at most of rank 1 and the processing across and parallel
to the image gradient are not independent. The processing
becomes independent if the image Hessian is used to compute
the diffusivity, even if it is not spatially averaged [20]. Computing the diffusivity from the structure tensor with spatial
averaging, however, results in a third alternative to trace-based
diffusion [22], which has a rank 2 diffusivity tensor and is
based on first order derivatives.
In the present work, the latter approach is modified by
considering the diffusion tensor as a function of the autocorrelation function. In particular, we use the Hessian, which
represents the curvature of the autocorrelation function, to
control the diffusion process. This is intuitively motivated by
the fact that the curvature (flat, parabolic, or elliptic) represents
the type of local structure (flat, linear, or two-dimensional)
to a large extend independently of intensity or texture. Flat
structures should be diffused isotropically, linear structures
only in one direction, and two-dimensional structures not at all.
Furthermore, we derive new discrete filter masks, apply a new
noise estimation method [23], and extend the experimental
evaluation using methods that take into account the visual
quality: the visual information fidelity (VIF) [24] and the
structural similarity index (SSIM) [25].
The paper is structured as follows. In Sect. II, we introduce some tensor notation and calculus and we give some
background on the structure tensor, anisotropic diffusion, and
adaptive filtering. In Sect. III, we derive the novel scheme
from the anisotropic diffusion equation and relate it to iterated
adaptive filtering, we propose concrete discrete filters for all
relevant steps and discuss practical aspects of the algorithm.
In Sect. IV, we present a number of standard denoising
experiments in order to compare our approach with methods
from the literature. Finally, we discuss the advantages and
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drawbacks of the proposed method in the concluding Sect. V.
II. T ENSORS , D IFFUSION , AND A DAPTIVE F ILTERING
In this section we introduce some notation, terms, and methods from the literature, which will be required in subsequent
sections.
A. Notation and Tensor Calculus
Despite the fact that tensors are coordinate-system independent algebraic entities, we restrict ourselves to matrix
representations of tensors in this paper. In what follows, (real)
vectors are denoted as bold letters and matrices as bold capital
letters. The scalar product between two column vectors a and
b is usually denoted by the matrix product of the transpose of
a and b: aT b. Tensors (of second order) will be often built
by outer products of vectors, e.g., T = abT . Since second
order tensors form a vector space, a scalar product (Frobenius
product) is defined, which reads for two tensors A and B
XX
hA|Bi = trace(ABT ) =
aij bij .
(1)
i
j
In most cases we will consider symmetric tensors, i.e.,
aij = aji and for these tensors the spectral theorem allows
a decomposition into real eigenvalues:
A = OΛOT = O diag(λi ) OT
(2)
where O is an orthogonal matrix formed by the eigenvectors of
A and Λ is the diagonal matrix containing the corresponding
(real) eigenvalues λi . Note that arbitrary powers of A can be
computed as
An = (OΛOT )n
=
OΛn OT = O diag(λni ) OT (3)
=
O diag(exp(k λi )) OT .
and due to linearity
exp(k A)
(4)
In many cases we deal with vector-valued or tensor-valued
functions in this paper, even if written as vectors or tensors.
The spatial variable, mostly denoted as x, is often omitted.
In this context, differential operators are based on the partial
derivatives with respect to the components of x and are
denoted using the nabla operator ∇, such that
∂b
(5)
∇b =
∂xi i
X ∂ai
∇T a = div(a) =
(6)
∂xi
i
!
X ∂aij
T
∇ A = div(A) =
.
(7)
∂xi
i
j
B. Structure Tensor
Tensor representations occur frequently in image processing, mostly for representing local orientation and the deviation from a simple signal model [6]. The gradient based
structure tensor can be derived as the solution of a leastsquares approach to fitting a simple signal model [26]. Another
alternative is to consider the structure tensor as an estimate of
the Hessian of the autocorrelation function of the signal [27].
Assuming ergodicity, the autocorrelation function of a 2D
image b(x) is given by correlating the image with itself
R = b ? b − b20
(8)
where b0 denotes the DC part of b. Since R is a symmetric
function, we might expand it as
1
R(x) = R(0) + xT HRx + O(|x|4 )
(9)
2
where H denotes the Hessian operator. With the power theorem we obtain (see [28], p. 317)
HR = E[∇b∇T b]
(10)
where E[·] denotes the expectation value. This expectation
value is usually estimated by locally weighted averaging the
outer product of the image gradient [29], [30]:
Z
J(x) = w(y)∇b(x − y)∇T b(x − y) dy .
(11)
Note that the gradient operators in the equation above are normally regularized by some low-pass filter, e.g., using Gaussian
derivatives. The local weight function w is also mostly chosen
as a Gaussian function.
C. Anisotropic Diffusion
We will not give a detailed introduction to anisotropic diffusion, the reader is referred to, e.g., the work of Weickert [31].
The diffusion scheme is defined as an evolution equation of
an image b(x) over time t as:
∂b
= div(D(J)∇b)
(12)
∂t
where D(J) is the diffusion tensor, which is computed by
modifying the eigenvalues of the structure tensor J. Applying
the product rule, the divergence term is split into two parts
(cf. (7))
div(D(J)∇b) = div(D(J))∇b + trace(D(J)Hb) .
(13)
If the structure tensor is computed by spatial averaging (11),
the divergence of the diffusion tensor is non-zero in general
and needs to be estimated. If the tensor is computed by
averaging over the different dimensions of a vector-valued
image, i.e., no spatial averaging is applied, the divergence
term can be converted into a trace-based term [21], [20].
Thus, by modifying D(J), the evolution equation can be
expressed by a single trace-based term, i.e., the divergence
of the diffusion tensor is eliminated. However, according to
our knowledge, this elimination is impossible if the structure
tensor is computed by spatial averaging.
When implementing anisotropic diffusion, an iterative algorithm that updates the input image successively has to
be implemented. In each time-step, the diffusion tensor and
the image gradient have to be estimated. If the evolution
is divergence-based, another numerical approximation of a
derivative operator is required, which is applied to the product
of diffusion tensor and gradient. Hence, five sequential operations are required: image gradients, point-wise products, local
FELSBERG: AUTOCORRELATION-DRIVEN DIFFUSION FILTERING
3
averaging, point-wise product, and gradient. If the evolution is
reformulated as a trace-based approach, only four sequential
operations are required, since the final gradient is replaced
with a second image derivative which can be computed in
parallel to the diffusion tensor.
D. Adaptive Filtering
Adaptive filtering is a more general and actually earlier published variant of steerable filters [32], developed by Knuttson
et al. [33], [34]. The main idea is to compose a spatially
variant filter kernel by linear combinations of shift invariant
kernels. The linear coefficients are locally estimated by an
orientation-dependent scheme. A comprehensible formulation
of the method can be found in [6], Chapt. 10. Adopting the
notation above, the filter is composed as
X
hadapt = hLP +
hC(J)|Ñk ihHP,k
(14)
k
where hHP,k is an orientation selective high-pass filter with
orientation nk , Ñk is the dual tensor for the orientation tensor
nk nTk and C is the control tensor. The high-pass filter with
orientation nk is defined in the Fourier domain by a polar
separable filter with radial component ρ(|u|) and an angular
component D(u/|u|) = (nTk u)2 /|u|2 . Consider for instance
the case ρ(|u|) = |u|2 such that
HHP,k = (nTk u)2
(15)
p
p
with n1 = (1, 0)T , n2 = (0, 1)T , and n3 = ( 1/2, 1/2)T .
The resulting frequency responses are HHP,1 = u21 , HHP,2 =
u22 , and HHP,3 = u21 /2 + u1 u2 + u22 /2.
The corresponding orientation tensors are given as
1
0
0
0
1/2
1/2
N1 =
N2 =
N3 =
0
0
0
1
1/2
1/2
The dual tensors can be computed via the isometric vector
representation as
0 − 21
0
1
1 − 12
Ñ2 =
Ñ3 =
.
Ñ1 =
1
0
0
1
− 12
− 12
Finally, in our example, the rightmost term in (14) is proportional to the negative Hessian since
X
Ñk HHP,k = uuT .
(16)
k
The tensor controlled adaptive filter scheme is often applied in
an iterated way to achieve good denoising results [6], Chapt.
10. By selecting the high-pass filter in the way we have done
in the example above, but with a very small constant multiplier
0 < γ << 1 and selecting the low-pass filter appropriately as
HLP = 1 − γρ(|u|) = 1 − γ|u|2 ,
(17)
it is possible to show that the iterated adaptive filtering
implements a numerical scheme for autocorrelation-driven
anisotropic diffusion, see Sect. III-B.
III. AUTOCORRELATION -D RIVEN D IFFUSION F ILTERING
This section starts with defining autocorrelation-driven diffusion filtering in the continuous domain, where the autocorrelation function estimate is inserted after computing the
divergence, resulting in a purely trace-based formulation. This
continuous equation is then discretized in the temporal domain
and an equivalence to iterated adaptive filtering is found.
Finally, the scheme is fully discretized, its stability is analyzed,
and practical aspects are discussed.
A. Continuous Autocorrelation-Driven Diffusion Filtering
Many schemes for tensor-driven anisotropic diffusion use
the structure tensor for controlling the diffusion process.
Interpreting the structure tensor as an estimate of the autocorrelation function, cf. Sect. II-B (9) to (11), one can go
one step back and reformulate anisotropic diffusion with a
diffusion tensor that depends on the autocorrelation function
instead:
Definition 1: The autocorrelation-driven anisotropic diffusion is defined by the evolution
∂b
= div(B(R)∇b) ,
(18)
∂t
where B(R) is the diffusion tensor determined by the autocorrelation function R at the current spatial position.
This definition is particularly sensible from a statistical point
of view if we consider the ergodicity assumption which is
essential for denoising by weighted spatial averaging. If the
image signal is ergodic in a strict sense, all statistical moments
are stationary and thus the autocorrelation is constant, resulting
in unweighted averaging. If the image signal is structured,
its statistical moments are not stationary, and thus, it is not
ergodic and should not be averaged spatially. The autocorrelation function becomes highly peaked in all directions in
this case. If the image signal is ergodic in one direction,
but structured in the orthogonal direction (e.g., edge or line
structure), averaging should only be performed orthogonal to
the structure. The autocorrelation function is peaked along the
structured direction and flat orthogonal to it.
In this work, we consider anisotropic diffusion (12) as an
approximation to (18). We will now derive another approximation to (18) which is of fourth order and which always
guarantees an even autocorrelation function:
Theorem 1: Assume that J according to (11) is at least a
second order approximation to HR and B is continuously
differentiable. Then
∂b
= hD(J)|Hbi
(19)
∂t
is a fourth order approximation to (18) for a suitable function
D.
Proof: For the diffusivity B(R), a fourth order approximation is given as
1
B(R) ≈ B(R(0)) + xT HRx B0 (R(0)) .
(20)
2
Since J is at least a second order approximation of HR,
1
.
D(J) = B(R(0)) + xT Jx B0 (R(0))
(21)
2
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is also a fourth order approximation of B(R). If we now apply
the product rule to (18), we obtain with (1)
div(B(R)∇b) = div(B(R))∇b + hB(R)|Hbi .
(22)
For the first term, we obtain
div(B(R)) = B0 ∇R
(23)
which vanishes in the origin since R is an even function.
Hence,
div(B(R)∇b) = hB(R)|Hbi ,
(24)
and plugging in (21) we obtain that (19) is a fourth order
approximation to (18).
If we plug the autocorrelation approximation (21) into (18)
before computing the divergence, we obtain the standard
formulation of anisotropic diffusion (12). The only difference
to (19) is that the divergence of the diffusion tensor does not
vanish in general:
div(D(J)) 6= 0 .
However, this implies that the autocorrelation function contains a first order term ∂R
∂x , leading to a third order approximation to (18). Note that this result is just on the approximation
order and assumes that (18) is our model instead of (12).
B. A Time-Discrete Scheme for Autocorrelation-Driven
Anisotropic Diffusion
In order to obtain a filtering method, the continuous differential equation (19) needs to be discretized in the spatial
and temporal domain. We start with discretizing the temporal
domain, since we aim at relating diffusion filtering to adaptive
filtering, which has been introduced with continuous notation
in Sect. II-D. Using a standard explicit scheme for time
discretization, we obtain from (19)
bt+1 = bt + γhD(J)|Hbt i
(25)
where we use γ = 0.05 throughout this paper. The upper limit
for γ depends on the spatial discretization and is discussed in
Sect. III-D.
For the iterative filtering (25), we obtain
Theorem 2: Autocorrelation-driven diffusion filtering according to (25) and the particular case of iterated adaptive
filtering using (15) are equivalent.
Proof: As it has been shown in the example in Sect. II-D,
the Hessian can be written in terms of P
a set of dual tensors
and corresponding filters (16). Hence, k Ñk hHP,k = −H
and plugging this into (25) gives
X
D
E
bt+1 = bt + γ Id − C(J) −
Ñk hHP,k bt
(26)
k
where we replaced D(J) = Id − C(J). Sorting terms gives
!
X
bt+1 = δ − γ(hHP,1 + hHP,2 ) +
hC(J)|Ñk iγhHP,k bt
k
(27)
which is equivalent to an adaptive filtering according to (14)
with constant multiplier γ and the low-pass filter δ−γ(hHP,1 +
hHP,2 ). This low-pass filter is the inverse Fourier transform
of (17).
C. A Fully Discrete Scheme for Autocorrelation-Driven
Anisotropic Diffusion
In order to derive a fully discrete scheme for (25), we
also discretize the spatial domain. Three filter (sets) have to
be designed: The Hessian, the derivatives in (11), and the
averaging kernel in (11). The latter is chosen to be a Gaussian
function with variance σ 2 = 1 throughout the paper. The
exact value for the variance is not crucial, as the experimental
results do not differ for small variations. The derivatives used
in the structure tensor (11) and the second derivatives used
to compute the Hessian are computed with filter masks of
minimum size. This is motivated by an observation made
in [35]: Reducing linear regularization, i.e., using smaller
filter masks, for the benefit of increased regularization of the
structure tensor improves the overall result. In order to obtain
small filter masks with good isotropy, we apply a similar
optimization strategy as in [36], but with a different objective
function. We start by assuming 3 × 3 filter masks, which gives
us one degree of freedom for the first derivatives (p ∈ [0; 0.25]
since we want to avoid extra zeros in the frequency response):
∂b
∂x1
∂b
∂x2
≈
≈
1
([p 1 − 2p p]T [−1 0 1]) ∗ b
2
1
([−1 0 1]T [p 1 − 2p p]) ∗ b .
2
(28)
(29)
A second order derivative approximation can also be computed
with a 3×3 filter, which we derive from one-sided finite differences. This guarantees frequency responses with zero DC level
and without extra zeros. Starting with the second derivative for
the horizontal coordinate, the 1D Laplace filter is obtained by
combining a left-sided and a right-sided difference:
∂2
≈ [−1 1 0] ∗ [0 − 1 1] = [1 − 2 1] .
∂x21
(30)
A second approximation without extra zeros is obtained by the
2D filters that average in the vertical direction with a 2-box
filter (see also [5]):

 

−1
1
0
0
0
0
∂2
1
−1
1
0 ∗ 0
−1
1
≈
∂x21
4
0
0
0
0
−1
1


1
−2
1
1
2
−4
2 .
=
(31)
4
1
−2
1
The actually used filters are the weighted average of both
approximations (30) and (31), which span the whole space
of zero DC 3 × 3 second derivatives without extra zeros in the
frequency domain:


q
−2q
q
∂2
1
4 − 2q
−8 + 4q
4 − 2q  (32)
≈
∂x21
4
q
−2q
q


q
4
−
2q
q
∂2
1
−2q
−8 + 4q
−2q 
≈
(33)
∂x22
4
q
4 − 2q
q
FELSBERG: AUTOCORRELATION-DRIVEN DIFFUSION FILTERING
The mixed derivatives corresponding to (30) and
identical:
   T

−1
1
0
−1
1
1   
∂2
0 ∗ 0
0
0
≈
=
∂x1 ∂x2
4
4
1
−1
0
1
5
(31) are

−1
0 .
1
(34)
The optimal filterset in terms pf p and q is obtained by minimizing the weighted angular error between two normalized
tensors in the Fourier domain:
1
H12
H1 H2
(35)
T1 =
H22
H12 + H22 H1 H2
1
H11 H12
, (36)
T2 = − p 2
2
2
H11 + H22 + 2H12 H12 H22
where u is the 2D frequency, Hk = Hk (u; p) are the Fourier
transforms of ∂x∂ k and Hkl = Hkl (u; q) are the Fourier
2
transforms of ∂x∂k ∂xl . The weighting function v = v(|u|) is
determined by the weight w in (11) and the statistics of natural
images (|u|−1 ). Hence, we minimize
Z
2
min v cos−1 (hT1 |T2 i) du .
(37)
p,q
The resulting parameters p and q are given in good approximation as p = 14 , i.e., a Sobel filter for the first derivative, and
q = 13
16 .
D. Stability of the Discrete Scheme
Conditions for the stability of discrete diffusion schemes
have been proposed in [37]. The fully discrete scheme is
rewritten in the form
bt+1 = Q(bt )bt
(38)
and the function Q(bt ) is tested for six criteria (D1)–(D6): continuity in its argument, symmetry, unit row sums, nonnegative
off-diagonals, irreducibility, and positive diagonal. In our case,
we derive a parametrized effective convolution kernel, which
can then be converted into the corresponding matrix operator
Q(bt ).1
First of all, we need to specify the diffusivity function D(J).
It is known from the literature that the diffusivity determines
the relation between non-linear diffusion and robust estimation [17], [38]. The width of the influence function in robust
estimation is more important than the exact shape [38], such
that we decided to choose an algebraically simple diffusivity
of exponential form:
1
(39)
D(J) = exp − J .
k
Due to (4), this corresponds to applying the corresponding
exponential mapping to the eigenvalues. Without loss of generality, we assume k = 1 for the reminder of this section.
1 This actually means that we assume an infinitely large or periodic discrete
image. For finite domains, Q(bt ) is no longer equivalent to a convolution
kernel.
Let λ1 > λ2 denote the eigenvalues of J and θ the angle
of the eigenvector corresponding to λ1 . Hence, using σk =
exp(−λk ) (note σ2 > σ1 )
d11 d12
D(J) =
(40)
d12 d22
σ1 cos2 θ + σ2 sin2 θ −(σ2 − σ1 ) cos θ sin θ
=
−(σ2 − σ1 ) cos θ sin θ σ1 sin2 θ + σ2 cos2 θ
and we obtain (using α = σ1 + σ2 = d11 + d22 )


αq + 2d12 4d22 − 2αq αq − 2d12
hD(J)|Hi = 4d11 − 2αq 4α(q − 2) 4d11 − 2αq 
αq − 2d12 4d22 − 2αq αq + 2d12
(41)
Finally, the filter that corresponds to Q(bt ) is given as δ(x) +
γhD(J)|Hi, cf. (25).
For Q(bt ), we immediately get continuity, symmetry, unit
row sums, and irreducibility. Positive diagonal entries (D6) are
obtained if
2
.
(42)
γ<
19
The off-diagonals are however not nonnegative (D4), choose
e.g. σ2 = 1, σ1 = 1/15 and θ = π/4, the upper left
element in (41) equals −1/15. Even though the nonnegativity
is violated, this does not imply instability of the scheme.
In particular, for q = 0, the proposed scheme is identical
to the explicit scheme [39], which is known to be conditional
stable [40]. If we restrict the stability analysis to locally simple
signals, we can draw conclusions concerning stability. For this
class of signals, the structure tensor J is of rank one. Hence,
σ1 1 and σ2 = 1. Assuming σ1 = 0, we can compute
the θ-intervals where the coefficients in (41) become negative.
These are (modulo π):
4d22 − 2αq
(43)
(0.15π; 0.35π) for
αq + 2d12
(44)
(0.28π; 0.72π) for
4d11 − 2αq
(45)
(0.65π; 0.85π) for
αq − 2d12
(46)
(−0.22π; 0.22π) for
These intervals are such that the angular distance to the
gradient orientation is maximized, i.e., we have a maximum
level of integration in these sectors and due to the vanishing
sums property, the resulting effective operator is nonnegative.
For instance, if we integrate along the columns (assuming
θ = 0) of (41), we obtain the weighted 1D Laplacian operator
σ1 [4, −8, 4]. It also worthwhile noticing that for the case
q = 0, the first (43) and the third sector (45) vanish, but the
second (44) and fourth (46) cover the full respective quadrant.
For q = 1, the first (43) and third sector (45) cover the full
respective quadrant and the second (44) and fourth sector (46)
vanish.
E. Practical Details
What we experienced as the most critical parameter of the
algorithm in order to obtain comparative results, is the width
of the diffusivity function (39) k. Too large values of k lead to
blurred results, i.e., structure is considered as noise, and too
small values lead to noisy results, since noise is considered
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as structure. The parameter that gives the statistically best
discrimination of structure and noise can be derived from a
null-hypothesis test and the resulting choice for k is [41]
k=
e−1 2
σ
e−2 e
(47)
where σe2 is the estimated noise variance according to the
method described in [23] with a minor modification. The
estimation is based on a mixture of two χ22 distributions,
which are fitted to the squared gradient magnitude distribution
using the EM algorithm. The minor modification that we have
applied here is, that we swap the two mixture components
during the iterations if the noise has a larger estimated variance
than the signal. Note further that the variance is given as twice
the mixture parameter µ of the χ22 distribution: σe2 = 2µ.
When implementing the diffusivity function (39), it appears
to be preferable to apply the exponential mapping directly on
the matrix instead of an eigenvalue manipulation. However, the
latter is faster since the eigenvalues can be extracted in closed
form [5], [26]. In principle this would also allow to combine
explicitly isotropic and anisotropic diffusion [42], but we chose
to apply the same scheme to edge-like images (e.g. the pepper
image) and fingerprint images since the achieved results were
satisfying in both cases. Our scheme shares similarities with
edge-enhancing diffusion, which explains good performance
on edge-like images. Good performance on line-like structures
might be explained by the missing odd order terms in the
scheme due to omitting the divergence of the diffusion tensor.
Finally, an often discussed parameter of anisotropic diffusion is the stopping time. We will not give a detailed consideration here. In our implementation the iterations are stopped as
soon as the image update is below a certain threshold. Based
on k, the threshold for the stopping criterion is empirically
set to k 1/5 /40 by looking at curves of maximum PSNR in
logarithmic scale. This is however only partly satisfying as
the rate of change might depend on the image structure and
not on the difference to a noise-free instance of the image. The
stopping time should rather be based on the estimated noise
level at each time instance. In practice, however, we observe
little difference of the two strategies, since the largest area of
the images is rather flat and the mean or median image update
is basically given by the amount of removed noise (which is
in itself directly related to the amount of noise).
IV. E XPERIMENTS
This section consists of two parts: the experiments and the
discussion of results.
A. Experimental Results
For the experiments, we used a popular set of images that
has already been used in earlier publications [12], [43], [44].
The evaluation criteria are, besides the PSNR, SSIM [25] and
VIF [24]. The PSNR does not reflect what kind of artifacts
are introduced, cf. Fig. 1, therefore, the other two criteria have
been included.
Two of the most successful methods in terms of PSNR
are due to Portilla et al. (GSM) [12] and Dabov et al.
Fig. 1. The PSNR does not reflect the type of residual error. Here: detail of
the pepper image (σ = 15, top left) for which the proposed method, GSM,
and EEDF yield about the same PSNR result. Top right: EEDF, bottom left:
GSM, and bottom right: proposed method.
(BM3D) [14], so that we compare our method with their implementations. Other methods as FoE [43] and K-SVD [13] are
respectively slightly inferior [15].2 Furthermore, we compare
our results to a standard implementation of anisotropic (edge
enhancing) diffusion filtering (EEDF) [3]. We evaluate each
image and noise level with ten different instances of noise,
clipped and quantized to eight bit integers. Each method is
fed with an estimated noise level and not with the ground
truth noise level of the added noise as done in the original
evaluations [12], [15]. The noise level is used as a parameter
in all four methods to determine the degree of smoothing.
Hence, results for GSM and BM3D differ from the literature,
since the original images already contain noise, resulting in a
higher noise level estimate.
For the smoothed result, PSNR, SSIM, and VIF values are
reported. Our method is entirely implemented in Matlab and
is about 6 times faster than GSM, which is a combination of
Matlab and C code. Our implementation is similar to the EEDF
implementation [3] and shares basically the same parameters.
Therefore, we use the same parameter settings for the EEDF,
2 The method in [15] seems to work slightly better than [14], but the code
for the latter was more straightforward to be used for denoising.
FELSBERG: AUTOCORRELATION-DRIVEN DIFFUSION FILTERING
7
cf. Section III-E. All our attempts to improve the results from
EEDF by varying the parameters have not lead to a systematic
increase of the quality.
The resulting PSNRs for the four methods are plotted in
Fig. 2 for two different images. Numerical results of the
house.png
50
proposed
EEDF
GSM
BM3D
45
40
TABLE I
PSNR S FOR THE PROPOSED METHOD
σ
1
2
5
10
15
20
25
50
75
100
Lena
47.8
42.5
37.5
33.4
31.6
30.4
29.7
25.8
22.8
20.5
Barb.
47.8
42.8
36.6
32.3
29.1
26.5
24.9
22.9
21.1
19.3
Boat
46.6
42.5
36.1
32.3
30.9
29.6
28.5
24.9
22.1
20.0
House
47.8
42.1
36.1
32.6
30.9
29.4
28.7
25.0
22.0
20.2
Pepper
48.1
42.8
36.9
33.2
30.7
29.6
28.9
24.1
21.4
19.4
Fingerp.
37.9
36.9
34.4
30.8
28.7
27.3
26.3
23.1
20.8
19.1
Flinst.
38.5
37.4
34.8
31.3
29.9
28.4
27.0
21.2
19.5
17.7
35
1
25
0.9
20
0.8
15
0.7
10
0.6
VIF
PSNR [dB]
house.png
30
5
0
0.5
0.4
1
2
5
10
15
20
25
50
75
100
m
0.3
fingerprint.png
50
0.2
proposed
EEDF
GSM
BM3D
45
40
proposed
EEDF
GSM
BM3D
0.1
0
1
2
5
10
15
20
25
50
75
100
25
50
75
100
m
35
1
25
0.9
20
0.8
15
0.7
10
0.6
VIF
PSNR [dB]
fingerprint.png
30
5
0
0.5
0.4
1
2
5
10
15
20
25
50
75
100
m
0.3
0.2
Fig. 2. PSNR results for the proposed method, EEDF, GSM, and BM3D for
the house image and the fingerprint image.
proposed method are summarized in Table I.
The resulting VIFs and SSIMs for the four methods are
plotted in Fig. 3 and Fig. 4, respectively. Numerical results
of the proposed method are summarized in Table II for the VIF
and in Table III for the SSIM. In order to allow a more systematic analysis of the performance, the differences between the
proposed method and the three competing methods are plotted
in Fig. 5.
B. Discussion of Results
As it can be seen from the PSNRs in Fig. 2 and the solid
line plots in Fig. 5 (top) for low noise levels, the proposed
proposed
EEDF
GSM
BM3D
0.1
0
1
2
5
10
15
20
m
Fig. 3. VIF results for the proposed method, EEDF, GSM, and BM3D for
the house image and the fingerprint image.
method works better than GSM and BM3D and slightly worse
than EEDF. For high noise levels, GSM and BM3D work
slightly better than the proposed method and EEDF works
worst. Integrated over the different noise levels, GSM works
only better than the proposed method for the house image,
BM3D works worse for boat, pepper, and fingerprint, and
8
TRANSACTIONS OF IMAGE PROCESSING, VOL. ..., NO. ..., ... 2011
TABLE III
SSIM SCORES FOR THE PROPOSED METHOD
house.png
1
σ
1
2
5
10
15
20
25
50
75
100
0.9
0.8
0.7
SSIM
0.6
0.5
0.4
Lena
1.00
0.99
0.98
0.94
0.90
0.87
0.84
0.67
0.53
0.43
Barb.
1.00
1.00
0.99
0.96
0.92
0.87
0.83
0.69
0.56
0.45
Boat
1.00
1.00
0.98
0.95
0.92
0.89
0.86
0.71
0.55
0.44
House
0.99
0.97
0.89
0.81
0.76
0.71
0.69
0.51
0.37
0.30
Pepper
0.99
0.98
0.94
0.89
0.83
0.81
0.81
0.59
0.45
0.35
Fingerp.
1.00
1.00
0.99
0.99
0.97
0.96
0.95
0.88
0.80
0.72
Flinst.
0.99
0.99
0.99
0.98
0.97
0.95
0.93
0.79
0.71
0.62
0.3
0.2
1
2
15
5
10
15
20
25
50
75
100
m
fingerprint.png
1
0.9
0.8
0.7
SSIM
0.6
10
6 PSNR / 100 6 VIF / 100 6 SSIM
0.1
0
difference as function of noiselevel
proposed
EEDF
GSM
BM3D
5
0
ï5
ï10
ï15
0.5
ï20
0.4
ï25
1
PSNR/GSM
PSNR/EEDF
PSNR/BM3D
SSIM/GSM
SSIM/EEDF
SSIM/BM3D
VIF/GSM
VIF/EEDF
VIF/BM3D
2
5
10
15
20
25
50
75
100
m
0.3
difference as function of image
0.2
1
2
20
5
10
15
20
25
50
75
100
m
Fig. 4. SSIM results for the proposed method, EEDF, GSM, and BM3D for
the house image and the fingerprint image.
TABLE II
VIF SCORES FOR THE PROPOSED METHOD
σ
1
2
5
10
15
20
25
50
75
100
Lena
1.00
1.00
0.99
0.98
0.96
0.94
0.91
0.76
0.65
0.55
Barb.
1.00
1.00
0.99
0.96
0.90
0.82
0.74
0.67
0.60
0.51
Boat
1.00
1.00
0.99
0.97
0.93
0.90
0.86
0.62
0.57
0.49
House
1.00
1.00
1.00
0.99
0.98
0.96
0.93
0.75
0.64
0.53
Pepper
1.00
1.00
0.99
0.98
0.97
0.94
0.89
0.72
0.61
0.50
Fingerp.
0.91
0.90
0.90
0.86
0.80
0.75
0.71
0.59
0.49
0.39
Flinst.
1.00
0.99
0.99
0.97
0.96
0.94
0.92
0.55
0.49
0.40
EEDF works only better for the fingerprint image, cf. Fig. 5
(bottom). The proposed method works as well as other recently
published methods, see e.g., [16] in comparison to Table I.
Note that the quite obvious deviation from Portilla’s original
evaluation for the fingerprint image at low noise levels is due to
using the estimated noise level instead of the added noise level
6 PSNR / 100 6 VIF / 100 6 SSIM
0.1
0
25
proposed
EEDF
GSM
BM3D
15
10
PSNR/GSM
PSNR/EEDF
PSNR/BM3D
SSIM/GSM
SSIM/EEDF
SSIM/BM3D
VIF/GSM
VIF/EEDF
VIF/BM3D
5
0
ï5
ï10
ï15
Lena
Barb.
Boat
House
image
Pepper
Fingerp.
Flinst.
Fig. 5. Comparison of results as a function of noise (top) and as a function of
the image (bottom). Positive values mean that the proposed method performs
better than the competing method.
ground-truth. The fingerprint image contains a lot of noise
even without added noise and hence the comparison for the
PSNR is made to a very noisy image. If using the added noise
level, the method reproduces the noise from the ’groundtruth’
image. The obvious interpretation of this observation must be
that GSM determines the amount of smoothing nearly entirely
FELSBERG: AUTOCORRELATION-DRIVEN DIFFUSION FILTERING
from the noise level parameter, which also leads to low-pass
effects, cf. also Fig. 1, bottom left. A similar dependency
on the noise parameter is observed for BM3D. The proposed
method is only mildly affected and EEDF is hardly affected
by changing the noise level parameter for the fingerprint case.
As it can be seen from the VIF scores in Fig. 3 and the dashdotted line plots in Fig. 5 (top) throughout all noise levels,
the proposed method works better than GSM and worse than
EEDF. BM3D is better than the proposed method in a very few
cases. The same order of results is obtained if the results are
integrated over the different noise levels, cf. Fig. 5 (bottom).
As it can be seen from the SSIM scores in Fig. 4 and
the dashed line plots in Fig. 5 (top) for low noise levels,
the proposed method works better than GSM and BM3D.
For high noise levels, GSM and BM3D work better than
the proposed method. The proposed method works better
than EEDF throughout, but with an increasing difference
with increasing noise level. Integrated over the different noise
levels, GSM and BM3D work better than the proposed one on
all images except for the fingerprint image and the proposed
method works better than EEDF for all images, cf. Fig. 5
(bottom). The proposed method works better than the fields
of expert except for the three highest noise levels, the house
image, and medium to high level noise in the pepper image,
cf. [16] in comparison to Table III.
The proposed method is significantly faster than GSM and
EEDF (measured on a 3GHz Intel Core 2 Duo). Furthermore, it
is extremely simple to implement and due to the low number
of sequential steps in each iteration well suited for a GPU
implementation. First performance evaluations on the GPU
(own framework within DirectX on an Nvidea 8800GTX)
allowed us to run 50 iterations in video real-time on PAL
image size (720 × 576). 50 iterations are sufficient for all
considered noise levels. The method can be made even faster
by not updating the diffusion tensor in every iteration, but only
in every second or third.
V. C ONCLUSION
We proposed a new approach to define tensor driven diffusion based on the autocorrelation function. We show that our
scheme is equivalent to a special case of adaptive filtering.
We propose an optimized filter setup for the computation
and compare the method to GSM, BM3D, and EEDF using
three quality measures, PSNR, VIF, and SSIM. The result of
the comparison is ambivalent in the way that the ranking of
methods depends on the choice of the quality measure, the
noise level, and the kind of image. Still, we observe that
BM3D is always better than GSM and that our method is
superior to GSM and BM3D for low-noise cases and images
with many line structures. Compared to EEDF, the proposed
method is clearly superior, except for the VIF scores. Visually,
the diffusion results (EEDF and proposed method) contain
fewer artifacts than the GSM results. The proposed scheme
is easy to implement and fast to execute. Hence, we believe
that our scheme is a good choice in many practical denoising
tasks.
9
ACKNOWLEDGMENT
I would like to thank Dr. Hanno Scharr and Kai Krajsek for
discussions on the topic and pointing me to references with
relevant work. I further thank Johan Hedborg for running the
evaluation on the GPU.
This research has received funding from the EC’s 7th
Framework Programme (FP7/2007-2013), grant agreements
21578 (DIPLECS) and 247947 (GARNICS), from ELLIIT,
the Strategic Area for ICT research funded by the Swedish
Government, and from the The Swedish Research Council
through a grant for the project Non-linear adaptive color
image processing.
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TRANSACTIONS OF IMAGE PROCESSING, VOL. ..., NO. ..., ... 2011
Michael Felsberg received the Diploma degree in
1998 and the PhD degree in 2002 (summa cum
laude, awarded ’Fakultätspreis 2002’) in engineering
from Christian-Albrechts-University Kiel, Germany.
During his PhD studies, he held scholarships from
the DFG and the German National Merit Foundation.
He became associate professor at Linköping University, Sweden, in 2004, docent in 2005, and full
professor of computer vision and head of division
in 2008. His research interests include multidimensional signal theory, image processing, low-level
computer vision, and signal-processing based approaches to probabilistic
representations and learning. He is coordinator of the Seventh Framework
EU Programme Project DIPLECS. He has published more than 90 conference
papers, journal articles, and book contributions on image processing, computer
vision, and pattern recognition. He received paper awards from the German
Pattern Recognition Society (DAGM) in 2000 and 2004 and from the Swedish
Society for Automated Image Analysis (SSBA) in 2007 and 2010. He received
the Olympus Award in 2005. He is vice-president of the SSBA.
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