1 s2.0 S147466701642210X Main
1 s2.0 S147466701642210X Main
1 s2.0 S147466701642210X Main
Keywords: Medical imaging and processing; Biomedical and medical image processing and
systems
ing them vulnerable to image under- and over-smoothing. where Ist is the intensity of a pixel s from image I at instant
t, λ is a scalar related to the diffusion rate, γ is a positive
We present a quantitative analysis describing ADF limita-
constant selected according to the desired smoothing level,
tions and a novel framework based on both the strongest
ηs stands for the set of adjacent pixels of s, g(·) is an
edges and on planar regions of the image, in order to t
ESF, and ∇Is,p is the magnitude of the image directional
set ADF parameters optimally. The evaluation comprises
gradient from pixel s to p at instant t. The directional
magnetic resonance (MR) images with different acquisition t
gradient ∇Is,p can be approximated by Ipt −Ist . To simplify
protocols. t
the notation, we will replace ∇Is,p with x whenever pixel
⋆ The authors thanks to Fapesp (Jovem Pesquisador 11/08573-4), information and the iteration number are irrelevant to the
CAPES, and CNPQ (486988/2013-9) for the financial support. context.
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19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014
(a) (b)
(a) (b)
(c) (d)
Fig. 4. (a) Sagittal slice of synthetic MR image of the brain
with noise intensity of 5% of the maximum intensity,
and the resultant images after applying ADF with (b)
(c) (d)
5, (c) 50, and (d) 200 iterations, fixing γ at 150.
Fig. 2. Axial slice of input synthetic MR images of the that both alternatives generate good results, depending on
human brain exposed to (a) 3% and (b) 5% of noise the post-processing objectives.
level; Best filtering results for (c) 3% and (d) 5% of
noise level, respectively, after 200 iterations with fixed
γ parameter manually set.
(a)
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19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014
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19th IFAC World Congress
Cape Town, South Africa. August 24-29, 2014
to the ADF limitations. Therefore, given a conservative- 0.16γλ , using ESF of Equation 4. (8)
ness parameter c ∈ [0, 1], γ is set by Equation 5 at each
As x = ±γ, these reductions in the gradient are the same
iteration:
reduction employed to γ value of the next iteration. Em-
pirical experiments showed that the estimated γ reduction
γF , γF ≤ γE ,
γ= (5) is very similar to the γ computed from γE and γF at each
c(γF − γE ), otherwise.
iteration with c = 1.0.
The closer we set c to 0, the more conservative the ADF
will be.
4.4 Stopping Criteria
120
Planar Region
110 We cannot rely on F as an stopping criteria, since a more
Standard Deviation
635
630
625 We used two different datasets in our experiments: BWP
620
615
with noise levels 0%, 1%, 3%, 5%, 7%, and 9% of the
610 maximum image intensity; and a real brain MR image
605 dataset (RBI) containing 19 1.5 Tesla and 10 3.0 Tesla
600 images.
595
0 200 400 600 800 1000 In the worse case when very poor SNRs was present, the
γ Value
search for initial γE and γF values took 9 ADF iterations
(b) each, and 7 more iterations were required for the actual
filtering process, totalling 25 iterations.
Fig. 8. Plots of the standard deviation of regions (a) F
and (b) E after the first iteration of ADF versus the All ESF produced satisfactory results under the proposed
employed γ parameter in a synthetic MR images with framework. Nevertheless, we lay emphasis here on the
noise level 9% of the maximum image intensity. best results obtained while employing Equation 4 due to
its faster descent behavior after achieving the maximum
intensity.
4.3 Iterative Gamma(γ) Updating
Images from BWP dataset required more or less ADF
The initial γ will be utilized at the first ADF iteration. iterations based on the noise level. The 9% noise level
It would be tempting to utilize the same strategy for image is ideal to show the behavior of the ADF with
the next ADF iterations. Nevertheless, this is a very respect to the conservativeness parameter c. Figure 9
time consuming procedure. As presented in Section 5, shows the filtering results, with several different c values.
it takes more ADF iterations to compute each of γE We can note that for smaller c values, the edges are
and γF parameters than to actually filter the image. An sharper, but the strongest noise pixels remain.
alternative is to compute the expected γ reduction based RBI dataset had distinct results for 1.5 Tesla and 3.0 Tesla
on the highest gradient noise pixels affected in the current images. 3.0 Tesla required just one ADF iteration, while
iteration. 1.5 Tesla could not be filtered without affecting the image
The maximum gradient value of Equation 1 using Equa- edges. Figure 10 shows the result for a 1.5 Tesla image,
tions 2 to 4 occurs for x = ±γ. As mentioned in Sec- with several different c values. We can state the same
tion 3, edge pixels probably have more adjacent pixels conclusions pointed by BWP dataset, showing that the
with similar intensity than noise pixels. We can reasonably proposed framework is appropriated for real MR images
state that most of edge pixels have more than 1/4 of of any acquisition protocol.
its neighbors with similar intensity, and that most of the Comparing to other ADF extensions, the proposed method-
noise pixels have less than 1/4. Then, we have a gradient ology with c = 1.0 will generate a very similar result
reduction of: to the work in Tsiotsios and Petrou [2013]. The result
of employing the methodology in Black et al. [1998] is
0.25e−0.5 γλ , using ESF of Equation 2, (6) more blurred than the proposed methodology with c = 1.0
(see Figure 11). None of the previous works was able to
0.125γλ , using ESF of Equation 3, (7) generate conservative results as proposed here.
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