Key words

1 Introduction

Mapping of the transverse relaxation times T2* and T2 (or relaxation rates R2* = 1/T2* and R2 = 1/T2) requires series of MR images with different echo times TE: The maps are obtained by fitting the exponential model curve S(TE) = S0 exp(−TE/T2(*)) to the signal intensities of each image pixel with increasing TE. This approach is inherently associated with decreasing signal-to-noise ratio (SNR) for the image volumes obtained for longer echo times. Similarly, there is a substantial signal attenuation and SNR reduction in diffusion-weighted MRI versus images acquired without diffusion-sensitizing gradients. Low SNR basically poses two general problems: First, it leads to a variability for the parameter estimates in the quantitative maps or derived model parameters. Second, it induces a systematic bias for the estimated parameter maps [1]. Please see Polzehl and Tabelow [2] for an elaboration of the issues and possible solutions.

In order to tackle the challenges caused by the variability in the estimated parameters due to low SNR a number of strategies can be followed. The obvious one is hardware related and includes improvements on the MR scanner side, such as dedicated RF coils or the use of cryogenically cooled RF probes [3]. Then, SNR is increased by reducing the variability of the measurement. Since the magnitude of the MR signal is directly related to the voxel volume [4], SNR increase can be achieved by reducing the spatial resolution of the image acquisition. This is obviously a very unfavorable solution. Alternatively, signal variability can be reduced by averaging of the signal over multiple acquisitions. This approach comes at the cost of prolonged acquisition times. Acceleration techniques like parallel imaging [5, 6] or compressed sensing [7] allow for more signal averaging within the same time span. Image postprocessing provides another alternative for reducing the variability of the data. Within this chapter, we specifically consider a class of methods that post-processes the acquired data, that is, smoothing. The basic idea is that MR images (like all images) possess a local spatial structure that is characterized by equal, or at least similar, noise-free signal values in vicinities of a location. Then, local averaging of the signal reduces the noise component of the signal. The simplest approach is the application of a Gaussian filter which performs a local averaging with weights defined by a (Gaussian) kernel function and a bandwidth. Due to its simple mathematical formulation it can be performed very fast on data volumes independent of the spatial dimensionality of the data. However, it comes with the blurring of the images, which is evident at edges between different tissues, especially hindering the analysis of fine structures within the volumes.

Alternatively, adaptive smoothing methods have been developed that aim to take the local properties of the specific data at hand into account. The specific methods are based on different methodologies, like anisotropic diffusion [8,9,10,11], nonlocal means [12], penalization techniques [13, 14] wavelet filtering [15], model-based methods [16, 17], the propagation-separation approach [18, 19], or other local techniques [20, 21]. Many of these techniques have been applied to diffusion-weighted MRI data but are also applicable for relaxometry measurements. Depending on the specific method and the input data at adaptive smoothing is performed on each image volume separately or uses the combined information from all available raw data. In contrast to nonadaptive filters such as the Gaussian filter, these methods typically require longer computation times and the choice of one or more smoothing parameters. Furthermore, in a similar way as the Gaussian filter induced blurring of images, adaptive filters might impose artifacts depending on their assumptions on the data like a cartoonesque appearance due to a step-function reconstruction of the images.

Here, we specifically consider the nonlocal means (NLM) filter [22, 23], which is of proven value for MRI. In contrast to defining the weights for averaging within the local vicinity of a location depending on the spatial distance only, it compares the image intensities between all voxels to define “nonlocal” weights for averaging intensities. Furthermore, it generally compares not only single-voxel intensities but small spatial patches of the data to better account for extended similar patterns. It is related to the bilateral filtering [24], where the local weights are refined by a factor related to local intensity differences; nonlocal means can thus be considered as a bilateral filter with infinite spatial bandwidth making it nonlocal. The filter can be applied to images in two dimensions but also in 3D or for multispectral [25] such as multiecho imaging techniques used in MR relaxometry.

Advantages of the application of the NLM filter are manifold: As a purely postprocessing method it does not require special hardware or sophisticated image acquisition techniques but can be applied offline. Many implementations and extensions of the original algorithm as open-source software are freely available. Examples are provided under: https://sites.google.com/site/pierrickcoupe/softwares/denoising-for-medical-imaging/dwi-denoising/dwi-denoising-software. Most of the implementations require MATLAB. Although smoothing parameters need to be tuned, default settings provide a very good starting point. Images processed with the NLM filter exhibit an improved SNR, show no extra blurring of the edges and preserve effective spatial resolution. Occasional and slight introduction of large-scale structures along the coordinate axis is a recognized limitation of the NLM filter because the comparison of spatial patches is in favor of these directions.

In this protocol we describe the application of a purpose-build implementation of the NLM filter for 2D MRI relaxometry data (see Note 1 for implementation details). The open source tool comes in two versions: (1) each echo time is filtered independently (2D-NLM) (2) similarities between image patches are estimated jointly for the complete stack of echo images (stackNLM). While the first is a straightforward block-wise implementation of the NLM filter, the second proposes a novel method exploiting the redundancy of information in MR relaxometry data (see Note 1 for implementation details). Both filters reduce the noise level in low SNR relaxometry data with version 2 offering slightly superior results (see Note 2 for a detailed example). Additionally, application of the MATLAB Image Processing toolbox function imnlmfilt is described. This function performs similar to 2D-NLM while being considerably faster. To complete the preprocessing of MRI relaxometry data, correction of the Rician noise bias is described.

This chapter is part of the book Pohlmann A, Niendorf T (eds) (2020) Preclinical MRI of the Kidney—Methods and Protocols. Springer, New York.

2 Materials

2.1 Software Requirements

The processing steps outlined in this protocol require the following:

  1. 1.

    The software development environments MATLAB® (The MathWorks, Natick, Massachusetts, USA; mathworks.com) or Octave (gnu.org/software/octave).

  2. 2.

    The MATLAB/Octave implementation of the NLM filter downloadable from github.com/LudgerS/MRIRelaxometryNLM

  3. 3.

    Optional: The MATLAB Image Processing Toolbox release R2018b or newer.

  4. 4.

    A MATLAB/Octave tool for noise bias correction downloadable from github.com/LudgerS/MRInoiseBiasCorrection. This tool includes detailed documentation to facilitate adaptation for other software development environments or use with multichannel coil data.

  5. 5.

    A tool for data import. This tool will depend on the data format:

    • DICOM data—MATLAB contains a build in function to import dicom data: dicomread. For Octave support is offered by the following package octave.sourceforge.io/dicom/.

    • Bruker data—Bruker also offers a toolbox to import data into MATLAB directly. To obtain Bruker’s pvtools contact Bruker software support (mri-software-support@bruker.com).

2.2 Data Requirements

  1. 1.

    The following protocol assumes that a pure noise scan was acquired to reliable estimate the data noise level.

  2. 2.

    See Note 3 for comments on noise scan acquisition.

3 Methods

3.1 Data Import

These steps should be executed for both the relaxometry data and the noise scan data. In the following sections it is assumed that the relaxometry data was stored in variable relaxData and the noise scan in variable noiseData.

3.1.1 DICOM Data

  1. 1.

    To import dicom data execute the command

    data = dicomread(‘filePath’);

    where ‘filePath’ is a string containing the relative or absolute path of the .dcm file.

3.1.2 Bruker Data

For a Linux or MacOS system, substitute the file separator \ by /.

  1. 1.

    Ensure that pvtools is on the search path:

    addpath(genpath(‘…\pvtools’))

    where ‘…\pvtools’ specifies the path of the pvtools folder.

  2. 2.

    Store the path to the scan you want to load in variable scanFolder.

  3. 3.

    Read the visu_pars parameter file:

    visuPars = readBrukerParamFile([scanFolder, '\visu_pars']);

  4. 4.

    Load the Paravision reconstruction:

    [data, ~] = readBruker2dseq([scanFolder, '\pdata\1\2dseq'], visuPars);

3.2 Correct Data Scaling

The data scaling needs to be corrected in some cases depending on the data format and vendor if the relaxometry data and noise scan were acquired with different numbers of averages. For example, in Bruker data, the individual averages are simply added instead of computing the mean (see Note 4 for a procedure to test the scaling convention). Assuming the noise scan was acquired with a single average and the relaxometry data with NA averages, use the following processing step:

  1. 1.

    relaxData = relaxData/NA

3.3 Noise Level Estimation

We assume the availability of a pure noise scan. In this case the complete noise data constitutes a background region. Assuming that the noise scan was acquired with a single average and the relaxometry data with NA averages, the noise level can be computed as

  1. 1.

    sigma = std(noiseData(:))/(0.6551*sqrt(NA));

The factor 0.6551 accounts for the different standard deviations in background and high SNR regions in MR magnitude images [1, 26, 27]. For data from multichannel coils see Note 5.

3.4 Application of the NLM Filter

3.4.1 Filtering of Individual Echoes with 2D-NLM

To apply the provided NLM filter implementation, use a loop over all echo times. It is assumed that the relaxometry images are two dimensional and that the echoes are stored along the third dimension of relaxData.

  1. 1.

    Set the parameters of the filter (see Note 6 for an explanation of choices)

    params.centerDistance = 1; % distance between individual blocks

    params.blockRadius = 1; % size of the blocks;

    params.searchRadius = 10; % determines the number of searched blocks

    params.beta = 0.5; % adjusts the filter strength

  2. 2.

    Prealocate memory for the filtered data

    filteredData = zeros(size(relaxData));

  3. 3.

    Apply the filter to every echo individually

    for ii = 1:size(relaxData, 3)

    filteredData(:,:,ii) = nlmFilter2D(relaxData(:,:,ii), sigma, params);

    end

3.4.2 Filtering Using stackNLM

  1. 1.

    Set the paramters of the filter (see Note 4 for an explanation of choices)

    params.centerDistance = 1; % distance between individual blocks

    params.blockRadius = 1; % size of the blocks;

    params.searchRadius = 10; % determines the number of searched blocks

    params.beta = 0.3; % adjusts the filter strength

  2. 2.

    Directly apply the filter to all echoes

    filteredData(:,:,ii) = stackNlmFilter(relaxData, sigma, params);

3.4.3 Filtering Using MATLAB’s Imnlmfilt

MATLAB’s imnlmfilt works on 2D images. Again a loop over the echo times is used:

  1. 1.

    Prealocate memory for the filtered data

    filteredData = zeros(size(relaxData));

  2. 2.

    Apply the filter to every echo individually

    for ii = 1:size(relaxData, 3)filteredData(:,:,ii) = imnlmfilt(relaxData(:,:,ii), ‘degreeOfSmoothing’, sigma);end

3.5 Bias Correction

To correct the Rician noise bias, the MRInoiseBiasCorrection toolbox is used. Importantly, the noise level sigma of the original data and not that of the filtered image needs to be given as an input parameter.

  1. 1.

    Ensure that the toolbox is on the search path

    addpath(‘…\MRInoiseBiasCorrection’)

    where ‘…\MRInoiseBiasCorrection’ specifies the path of the toolbox folder.

  2. 2.

    Apply the noise bias correction

    correctedImage = correctNoiseBias(imageData, sigma, 1);

    The last argument specifies the number of channels in the RF coil.

4 Notes

  1. 1.

    The provided tool follows the block-wise implementation of the NLM filter outlined by Coupé et al. for 3D data [28]. The noise level σ is expected to be provided by separate means such as the noise scan described in this protocol. The filtering parameter h is set to \( \sqrt{2\beta {\sigma}^2n} \), where n is the number of pixels in the compared blocks and β a parameter which can be controlled by the user (see Note 5). Noise bias correction was not integrated into the filter but is recommended as a subsequent step.

  2. 2.

    We present an example application to showcase the potential of the NLM filter for T2* relaxometry. For this purpose the kidney of an ex vivo rat phantom was scanned with a multiecho gradient-echo (MGE) sequence (TR = 50 ms, TE = 2.14 ms, 2.47 ms echo spacing, ten echoes, [38.2 × 48.5] mm2 FOV, 1 mm slice thickness, [202 × 256] image matrix, 80 repetitions, one average per repetition). The average of all 80 repetitions was used as a reference measurement. T2* maps were computed from unfiltered data and data filtered following the protocols of Subheading 3.4.1 (2D-NLM) and Subheading 3.4.2 (stackNLM).

    Figure 1 shows maps for a single repetition (NA = 1), the average of four repetitions (NA = 4) and the reference data with 80 averages. Maps showing the deviation from the reference (ΔT2*) and the magnitude deviation (|ΔT2*|) highlight the strongly reduced noise level in filtered images for NA = 1. No bias is introduced and edges are preserved. The filter working on the complete stack of echoes (stackNLM) is more robust regarding outliers. For NA = 4 only very little improvement is achieved due to the already good data quality, yet also no artifacts are introduced.

    The root-mean-square deviations (RMSD) of the T2* maps from the reference support this visual assessment (Fig. 2a). At low SNR (low number of averages) the filtered data outperform the maps computed without filtering. The improvement exceeds the gain achieved by doubling the measurement time. At higher SNR filtered and unfiltered data perform similarly. stackNLM consistently performs better than 2D-NLM. Figure 2b shows selected echoes for NA = 1 and the reference. Here the improved performance of stackNLM becomes obvious for data acquired with TE = 19.4 ms where signal is recovered with sharper detail.

    Figure 3 shows an in vivo example for both T2* and T2 data. Here little filtering is performed for the high SNR T2* data, whereas the T2 map is effectively denoised.

  3. 3.

    A pure noise scan is acquired by setting the excitation flip angle and reference power to zero so that no excitation occurs and pure noise is acquired. The receiver gain needs to be set identical to all other scans for which the noise level should be determined. The number of averages can be reduced to one, as the resulting change in noise level is easily compensated. Alternatively, the output of the RF power supply can be disconnected following the system adjustments.

  4. 4.

    To determine the employed convention of averaging, acquire high SNR phantom data with varying numbers of averages while keeping all other parameters fixed. Import the data as described in Subheading 3.1. If all scans show the same signal magnitude, the scaling step described in Subheading 3.2 should be omitted. However if for example doubling the number of averages also doubles the signal amplitude, rescaling should be performed.

  5. 5.

    Assuming a sum-of-squares reconstruction, the factor 0.6551 should be replaced by 0.6824, 0.6953, 0.7014 or 0.7043 for 2, 4, 8 or 16 receive element coil data [29].

  6. 6.

    The block-wise NLM filter compares overlapping square image patches or blocks. The larger the distance between block centers (params.centerDistance), the fewer blocks are in the image. Reducing the number of blocks will accelerate the filter but might lead to artifacts in the vicinity of contrast edges. params.blockRadius determines the size of the blocks and must always be larger than params.centerDistance to ensure overlap. The block is a square patch of area (params.centerDistance + 1)2. Increasing the size of blocks will improve performance in homogeneous areas but prevent the filter from finding similar blocks in areas with small features. params.searchRadius specifies the number of compared image blocks: each block is compared with the adjacent (params.searchRadius + 1)2 blocks to determine the corresponding filter weights. In the ideal case the algorithm would always search the complete image for similar blocks; however, the computational cost to do so would be prohibitive while the benefit diminishes fast with increasing search radius. params.beta specifies the parameter β introduced in Note 1. While theoretical considerations suggest a value close to 1, the optimal value of β is known to vary with the noise level [28]. Increasing the value of β will increase the smoothing effect. The proposed values of 0.5 for 2D-NLM and 0.3 for stackNLM were found to perform well over a wide range of noise levels.

Fig. 1
figure 1

Filtered and unfiltered kidney T2* maps acquired in an ex vivo rat phantom. The reference was computed from unfiltered data and NA denotes the number of averages. The second row shows the T2* deviation from the reference while the third row shows the magnitude of the T2* deviation

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Fig. 2
figure 2

(a) Root-mean-square deviation (RMSD) from the reference (NA = 80, unfiltered) for T2* maps computed from filtered and unfiltered data. (b) Image data for selected echo times (TE). The scaling is set individually for each row

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Fig. 3
figure 3

Filtered and unfiltered T2 and T2* maps acquired in a rat in vivo. In this experimental set-up (a healthy naive rat kidney being imaged using a dedicated 9.4T small animal MR system) the SNR is sufficiently high for the denoising to have only minor denoisng effect. The parameter β was set to 0.4 for the stackNLM filter to increase the smoothing effect

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