Informatica 40 (2016) 337–342 337
Adaptive Coherence-enhancing Diffusion Flow for Color Images
V. B. Surya Prasath
Computational Imaging and VisAnalysis (CIVA) Lab, Department of Computer Science
University of Missouri-Columbia, USA
E-mail: prasaths@missouri.edu
Keywords: coherence, structure tensor, anisotropic diffusion, image restoration
Received: July 7, 2016
Color image restoration is one of the fundamental problems in image processing pipelines. Variational reg-
ularization and diffusion partial differential equations (PDEs) are widely used in solving these low-level
image smoothing and noise removal problems. In this paper, we consider a new adaptive coherence en-
hancing diffusion (CED) filter which combines anisotropic diffusion and structure tensor derived diffusion
functions. By exploiting isotropic smoothing in homogeneous regions and anisotropic diffusion tensor
filtering in edges and corners we obtain a PDE flow which can removing noise while preserving impor-
tant image details. Compared to the original CED approach our proposed adaptive CED (ACED) obtains
stable smoothing results. Experimental results on synthetic and real color images show that the proposed
filter has good noise removal properties and quantitative measurements indicate it obtains better structure
preservation as well.
Povzetek: Predlagan je nov algoritem za obnavljanje barv slik.
1 Introduction
Image restoration is an important low-level image process-
ing which is still an active area of research in computer sci-
ence. Among a wide variety of image noise removal meth-
ods two important classes of techniques are variational reg-
ularization and partial differential equation (PDE) based fil-
ters [1]. Perona and Malik [2] proposed an anisotropic dif-
fusion filtering based on a nonlinear PDE for image denois-
ing and edge detection. Though the Perona-Malik (PM)
PDE obtained edge preserving restorations under noise, it
is known to create blocky artifacts in homogenous (flat re-
gions where the pixel values do not vary much) regions in
the resultant images.
Various modifications and adaptations of PM PDE in
particular and other PDEs in general have been proposed
in the last two decades. One of the important class of im-
proved PDE based filter is due to Weickert who provided
a unified theory of anisotropic diffusion [4]. The structure
tensor provided better orientation estimation and edge dis-
crimination for steering the diffusion process away from
image discontinuities and to make smoothing strong in ho-
mogenous regions. Weickert [5] proposed an elegant for-
mulation which handles coherence enhancement for color
images with tuning the eigenvalues of the structure tensor
in a controlled manner. Structure tensors provide a geo-
metric analysis of digital images via eigenvalues and vec-
tors and there have been applications in edge detection and
image denoising literature.
In this work, we base our new PDE based filtering ap-
proach on Weickert’s coherence-enhancing diffusion [5]
with an adaptive choice of diffusion functions for better
edge and corner preservation while smoothing out random
noise. For this, we utilize structure tensor eigenvalues
for controlling anisotropic smoothing according to geomet-
ric content of the images. This adaptive choice facilitates
isotropic diffusion in homogenous regions and anisotropic
diffusion near sharp edges, corners. Our proposed filter is
robust to noise and we conduct detailed experimental re-
sults on noisy synthetic and real images to prove the effec-
tiveness. Comparison results with related filters show that
we obtain better restoration results visually as well as based
on peak signal to noise ratio and structure similarity.
We conduct experimental results on synthetic and vari-
ous corrupted real color test images and test our method
against some related filters from the literature. Our exper-
iments show that both visually and quantitatively our pro-
posed adaptive approach obtained better restoration results.
2 Adaptive coherence enhancing
diffusion
2.1 Preliminaries
We start with the basic assumptions and notations of
coherence-enhancing diffusion filtering framework. Let
u0 : Ω ⊂ R2 → R be the input (possibly noisy) grayscale
image. Weickert [4] provided a unified tensor diffusion for-
mulation which is given by the following parabolic nonlin-
338 Informatica 40 (2016) 337–342 V.B.S. Prasath
(a) Input (b) CED [5] (c) Proposed ACED
Figure 1: Comparison of noise-free synthetic Square color image filtering with coherence-enhancing diffusion flows
at the same terminal time T = 25. (a) Inupt image, and (b) Original Weickert’s CED [5] (Eqn. (1) with (7)), and our
proposed ACED (Eqn. (1) with (8)). In (b) and (c) on the left we show the residue |u(x, T )− u0| which are enhanced for
better visualization. Our proposed method obtains stable salient edges of the central square and other texture regions are
grouped well.
ear PDE,
∂u(x, t)
∂t
= div (D(Jρ(∇uσ))∇u) , x ∈ Ω,
u(x, 0) = u0(x), x ∈ Ω,
u(x, t) = 0, x ∈ ∂Ω.
(1)
The resultant sequence of images {u(·, t)}Tt=0, for a finite
time T represents a nonlinear scale space. Here the diffu-
sion tensorD is dependent on the image information via the
structure tensor Jρ(∇uσ). Structure tensors encode local
image information with first order directional derivatives
and is given by,
Jρ(∇uσ)
=
(
Gρ ? (uσ)
2
x Gρ ? (uσ)x(uσ)y
Gρ ? (uσ)x(uσ)y Gρ ? (uσ)
2
y
)
(2)
where [(uσ)x, (uσ)y]T (XT denotes the transpose of vec-
tor/matrix X) is the gradient of uσ (pre-smoothed image
uσ = Gσ ? u), Gρ = (2πρ2)−1 exp (− |x|2 /2ρ2) is the
Gaussian kernel and ? denotes the convolution operation.
Let the eigenvalues and eigenvectors of the structure tensor
be (λ+, λ−), and (v+, v−) respectively. Weickert’s unified
tensor diffusion formulation is given by,
D = f+(λ+, λ−)v+vT+ + f−(λ+, λ−)v−vT−, (3)
where where f+, f− are the diffusivities perpendicular and
parallel to structure orientations. The eigenvectors of the
structure tensor Jρ matrix can be calculated as,
λ±
=
1
2
(
trace(Jρ)±
√
trace2(Jρ) + 4det(Jρ)
)
. (4)
For vector valued (multichannel) images u : Ω ⊂ R2 →
RN with u = (u1, u2, . . . , uN ) channels the PDE (1) can
be written using a common diffusion tensor,
∂ui(x, t)
∂t
= div
(
D(Jρ(∇uσ))∇ui
)
, x ∈ Ω,
u(x, 0) = u0(x), x ∈ Ω,
u(x, t) = 0, x ∈ ∂Ω.
(5)
For vectorial images the common structure tensor is given
by,
Jρ(∇uσ) =
N∑
i=1
wi Jρ(∇uiσ), (6)
with
∑N
i=1 w
i = 1, and wi > 0 are the averaging factors.
Interpretation of this tensor for vectorial images in terms of
eigenvalues and eigenvectors carries over from grayscale
case, see [5] for more details. A simple choice is to chose
wi = 1/N for all i = 1, . . . , N representing all channels
have similar meaning, range and reliability. We restrict our-
selves here to color images (RGB, N=3), and the formula-
tion holds true for multispectral imagery as well.
Weickert [5] proposed the following particular choices
for steering smoothing for coherence enhancement diffu-
sion (CED),
f+ =
{
γ + (1− γ) exp (− α(λ+−λ−)2 ) if λ+ 6= λ−,
γ, else,
f− = γ. (7)
with α > 0 is known as the coherence factor (if the co-
herence is inferior to α the flux is increasing with the co-
herence while if the coherence is larger then α the flux de-
creases as the coherence grows), γ > 0 a small parameter
added to keep the tensor diffusion matrixD in Eqn. (3) pos-
itive definite. Note that (λ+−λ−)2 measures the coherence
within a window of scale ρ. This particular choice obtained
good diffusion results when the structures are oriented in
one particular direction, however can smooth out corners
and other singularities as multiple directional information
is lost, see Figure 1.
2.2 Adaptive coherence-enhancing diffusion
In this work, to control the filtering better and to preserve
image singularities better we chose the following adaptive
Adaptive Coherence-enhancing Diffusion Flow for Color Images Informatica 40 (2016) 337–342 339
(a) CED flow at T = 10, 100, 300
(b) ACED flow at T = 10, 100, 300
Figure 2: Comparison of coherence-enhancing diffusion flows at the same terminal times for StarryNight painting by
van Gogh (Saint-Rémy, 1889, The Museum of Modern Art, New York, USA) color image. Top row: original Weickert’s
CED [5] (Eqn. (5) with (7)). Bottom row: Our proposed ACED (Eqn. (5) with (8)) at terminal iterations T = 10, 100, 300.
Note the different effect of flows on long level lines.
diffusivities,
f+ = exp
(
−
λ2+
β1
)
f− =
(
1− exp
(
−
λ2+
β1
))
exp
(
−
λ2−
β2
)
(8)
With this choice of diffusivities we observe the following
salient points:
X If either of the eigenvalues λ+ or λ− is high the dif-
fusion is now in the direction of v+ or v−, which in
turn means at corners (where λ± is high) anisotropic
diffusion is applied.
X Original CED formulation’s diffusion oriented in the
direction of v+ is kept intact and the diffusivity f− is
now incorporates orientation direction v− (coherence
orientation).
X The parameters β1, β2 control the diffusivities along
v+, v−.
X In homogeneous (flat regions where the pixel values
do not vary) areas the diffusion is still isotropic.
The diffusion PDE in Eqn. (5) where the diffusion matrix
in Eqn. (3) given with this diffusivities (8) obtains adap-
tive coherence enhancing diffusion (ACED) for smoothing
color images with salient edges, corners better preserved as
we will see in the experimental results next.
3 Experimental results
3.1 Setup and parameters
The PDE based filters (CED) are implemented using the
implicit finite differences method with coherence parame-
ter α = 5 × 10−4, γ = 0.01 (to keep D positive definite),
pre-smoothing Gaussian standard deviations σ = 4, ρ = 1,
and step size ∆t = 0.24. The new parameters in our ACED
β1 = 20, and β = 20 are set in all the experiments reported
here.
3.2 Comparison results
Figure 1 shows a comparison of Weickert’s original
CED [5] (Eqn. (1) with (7)), and our proposed adaptive im-
provement ACED (Eqn. (1) with (8)) at the same terminal
time T = 25. We show the residue |u(x, T )− u0| to high-
light the amount of noise removed in both these methods.
As can be seen, our proposed ACED preserves the cen-
tral square’s edges through the diffusion flow and smaller
texture regions are grouped better than the original CED
340 Informatica 40 (2016) 337–342 V.B.S. Prasath
Image CED [5] Ours
Baboon‡ 0.6047 0.6087
Barbara‡ 0.7129 0.7964
Boat† 0.6786 0.7013
Car‡ 0.7214 0.7853
Couple† 0.7016 0.7020
F-16† 0.8699 0.8898
Girl1† 0.7081 0.7174
Girl2† 0.8210 0.8522
Girl3† 0.8020 0.8309
House† 0.7024 0.7812
IPI† 0.8335 0.8929
IPIC† 0.7899 0.8125
Lena‡ 0.7581 0.7824
Peppers‡ 0.7737 0.8026
Splash‡ 0.7938 0.8136
Tiffany‡ 0.7633 0.8107
Tree† 0.7099 0.7325
Table 1: Mean structural similarity (MSSIM) metric values
for results of various schemes with noise level σn = 30 for
standard test color images from USC-SIPI Miscellaneous
dataset (size † = 256× 256 and ‡ = 512× 512). MSSIM
value closer to one indicates the higher quality of the de-
noised image. The top result in each color test image is
indicated by boldface.
Image CED [5] Ours
Baboon‡ 20.48 20.53
Barbara‡ 24.59 25.94
Boat‡ 25.21 24.98
Car‡ 26.01 25.83
Couple† 27.18 26.97
F-16† 24.56 26.50
Girl1† 26.66 27.21
Girl2† 29.34 28.05
Girl3† 29.92 29.68
House† 28.95 28.39
IPI† 30.38 29.32
IPIC† 29.05 28.64
Lena‡ 28.56 27.89
Peppers‡ 28.61 28.03
Splash‡ 31.65 31.18
Tiffany‡ 29.67 28.99
Tree† 25.73 25.10
Table 2: PSNR (dB) values for results of various schemes
with noise level σn = 20 for standard images of size
† = 256× 256 (Noisy PSNR = 22.11) and ‡ = 512× 512
(Noisy PSNR = 22.09). Higher PSNR value indicate bet-
ter denoising result. The top result in each color test image
is indicated by boldface.
(a) Noise-free
(b) Noisy
Figure 3: Comparison of (a) noise-free smoothing and
(b) noisy Baboon color image filtering with coherence-
enhancing diffusion (CED) flows at the same terminal
time T = 25. Left column - original Weickert’s
CED [5] (Eqn. (1) with (7)), right column - Our pro-
posed ACED (Eqn. (1) with (8)). Restored image, and
residue |u(x, T )− u0| (enhanced for better visualization)
are shown in each case. Our proposed method obtains bet-
ter noise removal and salient edges are preserved well. Bet-
ter viewed online and zoomed-in.
formulation.
To show visually the qualitative differences of the flows
we utilize the StarryNight, a painting by van Gogh
Adaptive Coherence-enhancing Diffusion Flow for Color Images Informatica 40 (2016) 337–342 341
(a) Day
(b) Night
Figure 4: We obtain interesting non-realistic photo realistic results with our proposed ACED. We show two examples (a)
daylight, and (b) night lights. Better viewed online.
(Saint-Rémy, 1889, Courtesy of The Museum of Modern
Art, NY, USA) color image of size 606 × 480. Figure 2
shows CED and our proposed ACED at iterations T = 10,
100, and 300. As can be seen, we obtain two different be-
haviors with respect to the coherency of long level lines.
CED obtains a long flowing structures whereas our ACED
obtains long lines interspersed with small flowing lines in-
side big structures. This property shows that our adaptive
choice of diffusivities (8) helps the flow retain corners and
singularities better than the original (7).
Next in Figure 3 we compare CED flows on noise-
free and noisy (additive Gaussian noise of standard de-
viation σn = 30 added in all three channels indepen-
dently) Baboon color image of size 512×512. Figure 3(a)
shows comparison on noise-free image and the correspond-
ing CED, proposed ACED results at the iteration T = 25.
Our proposed ACED obtains better coherency as can be
seen by mouth and surrounding whiskers. A similar visual
analysis shows in noisy case, Figure 3(b), indicate we ob-
tain better noise removal while maintaining all the salient
edges and thin linear structures. These are further corrob-
orated by the corresponding residue images showing how
342 Informatica 40 (2016) 337–342 V.B.S. Prasath
much of structure and random noise are removed.
We note that for a fair comparison to the original CED
formulation we kept all the parameters in our proposed
ACED the same including the terminal time T of the cor-
responding PDE flows. The convergence result for the dis-
cretized versions, as iterations increases t → ∞, of both
CED and proposed ACED are the same and we defer the
discussion of deeper theoretical results of the correspond-
ing nonlinear PDEs for future.
To quantitatively compare the noise removal and struc-
ture preservation we use two standard error metrics utilized
widely in the image processing literature. Peak signal to
noise ratio (PSNR) is given by,
PSNR = 20 ∗ log10
(
3× umax√
MSE
)
dB,
where MSE = (mn)−1
∑∑
(u− uO)2, mean squared er-
ror, with uO is the original (noise free) image, m × n de-
notes the image size, umax denotes the maximum value,
for example in 8-bit images umax = 255. A difference of
0.5 dB can be identified visually. Mean structural similar-
ity (MSSIM) index is in the range [0, 1] and is known to
be a better error metric than traditional signal to noise ra-
tio. It is the mean value of the structural similarity (SSIM)
metric [6]. We use the default parameters for SSIM and
the MATLAB code is available online1. The SSIM is cal-
culated between two windows ω1 and ω2 of common size
N ×N , and is given by,
SSIM(ω1, ω2) =
(2µω1µω2 + c1)(2σω1ω2 + c2)
(µ2ω1 + µ
2
ω2 + c1)(σ
2
ω1 + σ
2
ω2 + c2)
,
where µωi the average of ωi, σ
2
ωi the variance of ωi, σω1ω2
the covariance, and c1, c2 stabilization parameters. The
MSSIM value near 1 implies the optimal denoising capa-
bility of a method and we used the default parameters.
Table 1 and Table 2 show PSNR (dB) and MSSIM val-
ues for CED and our proposed ACED methods compared
on some standard color test images which are synthetically
perturbed by additive Gaussian noise of standard devia-
tion σn = 30. Though our proposed ACED is not the
top performing method in all the tested images in terms
of PSNR, we note that the purpose of adaptive CED is to
obtain smoothed images with coherent structures in tact.
Further, PSNR is known to be not the right metric in evalu-
ating the performance of denoising methods and MSSIM is
more apt. We remark that the optimal stopping time T > 0
for denoising is determined based on best possible MSSIM
value in these synthetically noise added cases.
Finally, we show in Figure 4 some smoothing results
from mobile phone imagery (12 mega-pixel). Figure 4(a)
shows a picture taken in day-light conditions with no flash
and our proposed ACED obtains flow like small structures
while keeping the bigger regions intact. A similar result is
observed in Figure 4(b) where a night time image is cap-
tured with an in-built flash. The smoothing property of our
flow provides visually pleasing results in both cases.
1https://ece.uwaterloo.ca/ z70wang/research/ssim/
4 Conclusions
In this work, we considered a new PDE based filter for
color image coherence enhancing smoothing and noise re-
moval. By a combination of anisotropic diffusion with
structure tensor driven adaptive functions, our method ob-
tains edge preserving smoothing results which result in bet-
ter noise removal capabilities. Experimental results on a
variety of noisy images indicate the potential of our pro-
posed approach and compared with other original coher-
ence enhancing diffusion filter we obtained better restora-
tion results as well. One of our important future work is
in extending the proposed method by incorporating other
adaptive diffusive regularizers and to handle mixed noise
removal [3] and consider multispectral imagery [7].
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