Nikkita Khattar^{1}, Mustapha Bouhrara^{1}, and Richard G. Spencer^{1}

Image denoising is used extensively for MR image post-processing. The nonlocal means (NLM) filter shows excellent noise reduction while preserving detail. NLM takes advantage of the structural redundancy in MR images by comparing local neighborhoods of voxels throughout the image, and estimating the intensity of an index voxel to be denoised through a weighted average of voxel intensities. However, this excludes patches that may be similar except for rotation or reflection, and therefore does not make full use of image redundancy. We introduce a multispectral implementation of NLM incorporating rotations and reflections, finding improved performance compared to conventional non-multispectral filtering.

Purpose

The multispectral nonlocal means filter (MS-NLM) incorporates information from MR data sets obtained with a variable acquisition parameter, and shows excellent edge preservation and noise reduction**Materials and Methods **

**The RR-NLM and RR-MS-NLM filters**

We consider MS image
sets deﬁned on a discrete grid **I**
describing the bounded 3D spatial domain spanned by the image, given by $$$S= \left\{S(i)|i\in I,S(i)\in ℝ^{k}\right\}$$$ where
*K* is the number of frames of the MS
data set. We deﬁne a frame *k* as a
particular image within the multispectral dataset obtained with a particular
value of a varying acquisition parameter such as echo time. For NLM, *K* = 1, while for MS-NLM, *K *> 1. The intensity of an
index voxel *i *is estimated as the
weighted mean of signal intensities calculated over all voxels *j* in a large search window of size *R* centered on *i* through^{2-3}:

$$A_{k}(i) =\frac{ \sum_j^R w(i,j)S_{k}(j)}{\sum_j^R w(i,j)}, [1]$$

where *w(i, j)* is the weight quantifying the similarity between two voxels *i* and *j* in a frame *k* and is given
by: $$$w(i,j) = exp(-\sum_{k=1}^K d_{k}(i,j)/Kh^{2})$$$ where $$$d_{k}(i,j) = \sum_{l=1}^L (n_{k,l}(i) - n_{k,l}(j))^{2}$$$ is
the Euclidean distance defining the local similarity between voxels *i* and *j* of the frame *k**. *$$$n_{k,l}(i)$$$ refers to the *l*^{th} signal intensity within a
local patch, * n*, centered on voxel

In the RR-MS-NLM, the local
patch **n**(*j*) is rotated and reflected for each rotation. These re-oriented patches are then compared
to the local patch, **n**(*i*), centered on *i*. For each rotation and
reflection, *t*, the weight *w _{t}*(

$$A_{k}(i) = \frac{\sum_{t=1}^T\sum_j^R w_{t}(i,j)S_{k}(j)}{\sum_{t=1}^T\sum_j^R w_{t}(i,j)}, [2]$$

where *T* is the total number of rotations and
reflections.

**Analysis**

We illustrate the RR-NLM
and RR-MS-NLM filters with quadrature rotations, that is, weights were
calculated for local patch rotations of 0°, 90°, 180°, and 270° and their
corresponding reflections (*i.e. T* = 8). More finely-spaced rotations may readily be
implemented. Results were compared to
those from NLM and MS-NLM, that is, without rotation or reflection. Analyses were performed on a synthetic
two-dimensional checkerboard image as well as on synthetic MS *T _{2}*-weighted brain datasets
obtained from BrainWeb.

Figure 1: Results of filtering a noisy checkerboard image using NLM and RR-NLM. Visual inspection of filtered images and error maps indicate the superior performance of RR-NLM.

Figure 2: Results of filtering noisy MS synthetic brain
datasets using NLM and RR-NLM. The full dataset was used for filtering, with
results shown for two values of TE. Visual
inspection of filtered images and error maps indicate that RR-NLM and RR-MS-NLM
out-perform their counterparts that do not incorporate RR. In addition, as expected, the use of MS
information leads to enhanced filtering performance,^{2} as seen by
comparing the results of MS-NLM with NLM, and the results of RR-MS-NLM with
RR-NLM. The best performance was seen
with RR-MS-NLM.

*1. Buades A, Coll B, Morel JM. A review
of image denoising algorithms, with a new one. SIAM Journal on Multiscale
Modeling and Simulation. 2005;4(2):490-530.
*

*
2. Manjón
JV, Robles M, Thacker NA. "Multispectral MRI de-noising using non-local
means", Proc. MIUA. 2007;41-45.
*

*
3. Bouhrara M,
Bonny JM, Ashinsky BG, et al. Noise estimation and reduction in
magnetic resonance imaging using a new multispectral nonlocal
maximum-likelihood filter. IEEE Trans Med Imaging. 2017;36:181-193. *

*
4. Kwan RKS, Evans AC, Pike GB. MRI
simulation-based evaluation of image-processing and classification methods.
IEEE T Med Imaging.1999; 8(11):1085-97.*