Response function estimation

A compulsory step in spherical deconvolution is deriving the ‘response function (RF)’, which is used as the kernel during the deconvolution step. For the white matter, this is the signal expected for a voxel containing a single, coherently-oriented fibre bundle. While some groups prefer to define this function using some ad-hoc template function (e.g. a diffusion tensor with empirical diffusivities), the MRtrix contributors are in preference of deriving this function directly from the image data, typically by averaging the diffusion signal from a set of empirically-determined ‘single-fibre (SF)’ voxels.

The process of estimating this function from the data is however non-trivial; there is no single unambiguous way in which this should be done. Earlier in the beta version of MRtrix3, we provided a command dwi2response that advertised automated determination of the response function, based on a published method with a few additional enhancements. Unfortunately user testing showed that this algorithm would not produce the desired result in a number of circumstances, and the available command-line options for altering its behaviour were not intuitive.

As a result, we are now instead providing dwi2response as a script. This was done for a few reasons. Firstly, it means that we can provide multiple different mechanisms / algorithms for response function estimation, all accessible within the one script, allowing users to experiment with different approaches. Secondly, because these Python scripts are more accessible to most users than C++ code, the algorithms themselves can be modified, or some may even choose to try devising their own heiristics for response function estimation. Thirdly, it reinforces the fact that there is unfortunately not a black-box, one-size-fits-all solution to this problem.

Here I will discuss some of the technical aspects of response function estimation, and describe the mechanisms by which the currently provided algorithms work. If however you are not interested in the nitty-gritty of this process, feel free to scroll to the bottom of the page.

Necessary steps

Looking at the process of response function estimation in full detail, there are four crucial steps. For each of these, I will also briefly mention the typical process used.

  1. Select those image voxels that are to be used when determining the response function - the ‘single-fibre mask’. Typical: Varies.
  2. Estimate the direction of the underlying fibres in each voxel. Typical: Often the diffusion tensor fit is still used for this purpose; though CSD itself can also be used as long as an initial response function estimate is available.
  3. Rotate the signal measured in each single-fibre voxel in such a way that the estimated fibre direction coincides with the z-axis. Typical: This may be done by rotating the diffusion gradient table according to the estimated fibre direction; or if the diffusion signal is converted to spherical harmonics, then a spherical convolution can be used.
  4. Combine these signals to produce a single response function. Typical: The m=0 terms of the spherical harmonic series (which are rotationally symmetric about the z-axis) are simply averaged across single-fibre voxels.

Of these steps, the first is the one that has caused the greatest difficulty, and is also the principle mechanism where the provided response function estimation algorithms vary. It will therefore be the primary focus of this document, though note that the other aspects of this process may also change with ongoing development.

dwi2response algorithms


In the previous version of MRtrix (‘0.2’), the following heuristic was suggested in the documentation for deriving the response function:

  • Erode a brain mask by a few voxels, to omit any voxels near the edge of the brain;
  • Select those voxels within the mask that have a Fractional Anisotropy (FA) of 0.7 or greater;
  • The estimate_response command would then be used to generate a response function, which would internally perform diffusion tensor estimation to get the fibre directions as well as the gradient reorientation.

Rather than this series of commands, dwi2response now provides a similar heuristic in-built as the fa algorithm. The primary difference is that by default, it will instead select the 300 voxels with the highest FA (though this can be modified at the command-line).

This algorithm is provided partly for nostalgic purposes, but it also highlights the range of possibilities for single-fbre voxel selection. One of the problems associated with this approach (over and above the feeling of uncleanliness from using the tensor model) is that in white matter regions close to CSF, Gibbs ringing can make the signal in b=0 images erroneously low, which causes an artificial increase in FA, and therefore such voxels get included in the single-fibre mask.


This algorithm is provided for cases where none of the available algorithms give adequate results, for deriving multi-shell multi-tissue response functions in cases where the voxel mask for each tissue must be defined manually, or for anyone who may find it useful if trying to devise their own mechanism for response function estimation. It requires manual definition of both the single-fibre voxel mask (or just a voxel mask for isotropic tissues); the fibre directions can also be provided manually if necessary (otherwise a tensor fit will be used).


This algorithm is intended for deriving multi-shell, multi-tissue response functions that are compatible with the new Multi-Shell Multi-Tissue (MSMT) CSD algorithm. The response function estimation algorithm is identical to that described in the manuscript: As long as EPI inhomogeneity field correction has been performed, and a tissue-segmented anatomical image (prepared in the 5TT format for ACT) is provided with good prior rigid-body alignment to the diffusion images, then these high-resolution tissue segmentations can be used to identify single-tissue voxels in the diffusion images. This algorithm is hard-wired to provide response functions for the most typical use case for MSMT CSD: An isotropic grey matter response, an anisotropic white matter response, and an isotropic CSF response; the output response functions are provided in the format expected by the msdwi2fod command. Those wishing to experiment with different multi-tissue response function configurations will need to use the manual algorithm (which will provide a multi-shell response function if the input DWI contains such data).

For reference, this algorithm operates as follows:

  1. Resample the 5TT segmented image to diffusion image space.
  2. For each of the three tissues (WM, GM, CSF), select those voxels that obey the following criteria:
  • The tissue partial volume fraction must be at least 0.95.
  • For GM and CSF, the FA must be no larger than 0.2.
  1. For WM, use the mask derived from step 2 as the initialisation to the tournier algorithm, to select single-fibre voxels.
  2. Derive a multi-shell response for each tissue for each of these three tissues. For GM and CSF, use lmax=0 for all shells.


This algorithm is a fairly accurate reimplementation of the approach proposed by Tax et al.. The operation of the algorithm can be summarized as follows:

  1. Initialise the response function using a relatively ‘fat’ profile, and the single-fibre mask using all brain voxels.
  2. Perform CSD in all single-fibre voxels.
  3. Exclude from the single-fibre voxel mask those voxels where the resulting FOD detects more than one discrete fibre population, e.g. using the ratio of the amplitudes of the first and second tallest peaks.
  4. Re-calculate the response function using the updated single-fibre voxel mask.
  5. Return to step 2, repeating until some termination criterion is achieved.

The following are the differences between the implementation in dwi2response and this manuscript:

  • Deriving the initial response function. In the manuscript, this is done using a tensor model with a low FA. I wasn’t fussed on this approach myself, in part because it’s difficult to get the correct intensity sscaling. Instead, the script examines the mean and standard deviation of the raw DWI volumes, and derives an initial lmax=4 response function based on these.
  • The mechanism used to identify the peaks of the FOD. In dwi2response, the FOD segmentation algorithm described in the SIFT paper (Appendix 2) is used to locate the FOD peaks. The alternative is to use the sh2peaks command, which uses a Newton search from 60 pre-defined directions to locate these peaks. In my experience, the latter is slower, and may fail to identify some FOD peaks because the seeding directions are not sufficiently dense.

For the sake of completeness, the following are further modifications that were made to the algorithm as part of the earlier dwi2response binary, but have been removed from the script as it is now provided:

  • Rather than using the ratio of amplitudes between the tallest and second-tallest peaks, this command instead looked at the ratio of the AFD of the largest FOD lobe, and the sum of the AFD of all other (positive) lobes in the voxel. Although this in some way makes more sense from a physical perspective (comparing the volume occupied by the primary fibre bundle to the volume of ‘everything else’), it’s possible that due to the noisy nature of the FODs at small amplitudes, this may have only introduced variance into the single-fibre voxel identification process. Therefore the script has reverted to the original & simpler peak amplitude ratio calculation.
  • A second, more stringent pass of SF voxel exclusion was performed, which introduced two more criteria that single-fibre voxels had to satisfy:
  • Dispersion: A measure of dispersion of an FOD lobe can be derived as the ratio between the integral (fibre volume) and the peak amplitude. As fibre dispersion increases, the FOD peak amplitude decreases, but the fibre volume is unaffected; therefore this ratio increases. The goal here was to explicitly exclude voxels from the single-fibre mask if significant orientation dispersion was observed; this can be taken into account somewhat by using the FOD peak amplitudes (as orientation dispersion will decrease the amplitude of the tallest peak), but from my initial experimentation I wanted something more stringent. However as before, given the difficulties that many users experienced with the dwi2response command, this algorithm in the new script errs on the side of simplicity, so this test is not performed.
  • Integral: By testing only the ratio of the tallest to second-tallest FOD peak amplitude, the absolute value of the peak amplitude is effectively ignored. This may or may not be considered problematic, for either small or large FOD amplitudes. If the peak amplitude / AFD is smaller than that of other voxels, it’s possible that this voxel experiences partial volume with CSF: this may satisfy the peak ratio requirement, but using such a voxel is not ideal in response function estimation as its noise level will be higher and the Rician noise bias will be different. Conversely, both in certain regions of the brain and in some pathologies, some voxels can appear where the AFD is much higher due to T2 shine-through; it may seem appealing to use such voxels in response function estimation as the SNR is higher, but as for the low-signal case, the Rician noise bias will be different to that in the rest of the brain. The previous dwi2response binary attempted to exclude such voxels by looking at the mean and standard deviation of AFD within the single-fibre mask, and excluding voxels above or below a certain threshold. As before, while this heuristic may or may not seem appropriate depending on your point of view, it has been excluded from the new dwi2response script to keep things as simple as possible.


Independently and in parallel, Donald also developed a newer method for response function estimation based on CSD itself; it was used in this manuscript. It bears some resemblance to the tax algorithm, but relies on a threshold on the number of voxels in the single-fibre mask, rather than the ratio between tallest and second-tallest peaks. The operation is as follows:

  1. Define an initial response function that is as sharp as possible (ideally a flat disk, but will be fatter due to spherical harmonic truncation). Limit this initial function to lmax=4, as this makes the FODs less noisy in the first iteration.
  2. Run CSD for all voxels within the mask (initially, this is the whole brain).
  3. Select the 300 ‘best’ single-fibre voxels. This is not precisely the ratio between tallest and second-tallest peaks; instead, the following equation is used, which also biases toward selection of voxels where the tallest FOD peak is larger: sqrt(|peak1|) * (1 - |peak2| / |peak1|)^2. Use these voxels to generate a new response fuction.
  4. Test to see if the selection of single-fibre voxels has changed; if not, the script is completed.
  5. Derive a mask of voxels to test in the next iteration. This is the top 3,000 voxels according to the equation above, and dilated by one voxel.
  6. Go back to step 2.

This approach appears to be giving reasonable results for the datasets on which it has been tested. However if you are involved in the processing of non-human brain images in particular, you may need to experiment with the number of single-fibre voxels as the white matter is typically smaller.


If this document appears far too long for your liking, or you’re not particularly interested in the details and just want to know what option to use so that you can continue with your processing, the following are our ‘cautious’ recommendations. However we emphasize that this script may not work flawlessly for all data; if it did, we wouldn’t be providing a script with so many options! Furthermore, these recommendations may change over time, so check in every now and then, and make sure to sign up to the new community forum.

  • If you’re processing single-shell data, the tournier algorithm appears to be fairly robust.
  • If you’re processing multi-shell data, and are able to perform EPI inhomogeneity distortion correction, msmt_5tt is currently the only fully-automated method for getting multi-shell multi-tissue response functions.