Elsevier

NeuroImage

Volume 44, Issue 1, 1 January 2009, Pages 83-98
NeuroImage

Threshold-free cluster enhancement: Addressing problems of smoothing, threshold dependence and localisation in cluster inference

https://doi.org/10.1016/j.neuroimage.2008.03.061Get rights and content

Abstract

Many image enhancement and thresholding techniques make use of spatial neighbourhood information to boost belief in extended areas of signal. The most common such approach in neuroimaging is cluster-based thresholding, which is often more sensitive than voxel-wise thresholding. However, a limitation is the need to define the initial cluster-forming threshold. This threshold is arbitrary, and yet its exact choice can have a large impact on the results, particularly at the lower (e.g., t, z < 4) cluster-forming thresholds frequently used. Furthermore, the amount of spatial pre-smoothing is also arbitrary (given that the expected signal extent is very rarely known in advance of the analysis). In the light of such problems, we propose a new method which attempts to keep the sensitivity benefits of cluster-based thresholding (and indeed the general concept of “clusters” of signal), while avoiding (or at least minimising) these problems. The method takes a raw statistic image and produces an output image in which the voxel-wise values represent the amount of cluster-like local spatial support. The method is thus referred to as “threshold-free cluster enhancement” (TFCE). We present the TFCE approach and discuss in detail ROC-based optimisation and comparisons with cluster-based and voxel-based thresholding. We find that TFCE gives generally better sensitivity than other methods over a wide range of test signal shapes and SNR values. We also show an example on a real imaging dataset, suggesting that TFCE does indeed provide not just improved sensitivity, but richer and more interpretable output than cluster-based thresholding.

Introduction

Many image enhancement and thresholding techniques make use of spatial neighbourhood information to boost belief in extended areas of signal. The motivation for considering neighbouring voxels is to increase sensitivity to regions of signal that are more spatially extended than the noise coherence. The most common such approach in neuroimaging is cluster-based thresholding, which is generally implemented as 2 stages, generally assuming that some spatial smoothing has previously been applied: first, threshold the raw statistic image (or parametric map) and identify resulting clusters of contiguous supra-threshold voxels, then calculate a p-value for each cluster on the basis of its size/mass (e.g., using Gaussian field theory or permutation testing).

Cluster-based thresholding is popular as it is often perceived to be more sensitive to finding true signal than voxel-wise thresholding; for example, cluster-based inference is more powerful when the spatial correlation length of signal exceeds that of noise and vice-versa for inference on the height of maxima (Friston et al., 1996). However, a limitation is the need to define the initial cluster-forming threshold (e.g., threshold the raw t-statistic image at t > 2.5). This threshold is arbitrary, and yet its exact choice can have a large impact on the results, particularly at the lower (e.g., t, z < 4) cluster-forming thresholds frequently used. It has not been possible to give more objective advice than “broader signals are best detected by low thresholds and sharp focal signals are best detected by high thresholds” (Friston et al., 1994). A second problem is that the initial hard thresholding introduces instability in the overall processing chain; small variations in the data around the threshold level can have a large effect on the final output.

A third problem, common also to simple voxel-based thresholding, is that the amount of spatial smoothing is itself arbitrary, given that the expected signal extent is very rarely known in advance of the analysis. Furthermore, one may well want to optimise sensitivity to different shapes and sizes of signal within one dataset simultaneously. An early attempt to address these issues was work on scale-space (Worsley et al., 1996a), but such approaches have not become widely-used, possibly because of the resulting increase in potential over-fitting of noise (and the associated increase in the number of multiple comparisons). Finally, a fourth problem is that it can be hard to directly interpret the meaning of (what may ideally be) separable sub-clusters or local maxima within very extended clusters (see the real data Example shown below, though see also the discussion regarding multi-level inference in Friston et al. (1996)).

In this paper we suggest a new method which attempts to keep the sensitivity benefits of cluster-based thresholding (and indeed the general concept of “clusters” of signal), while avoiding (or at least minimising) the problems listed above. The method takes a raw statistic image and produces an output image in which the voxel-wise values represent the amount of cluster-like local spatial support. The method is thus referred to as “threshold-free cluster-enhancement” (TFCE). It is simple (Fig. 1): each voxel's new value is given by the sum of the “scores” of all “supporting sections” underneath it; each section's score is simply its height h (raised to some power H) multiplied by its extent e (raised to some power E). The output value is therefore a weighted sum of the entire local clustered signal, without the need for a hard cluster-forming thresholding. For inference, the TFCE image can easily be turned into voxel-wise p-values (either uncorrected, or corrected for multiple comparisons across space) via permutation testing.

In summary, to optimise the detection of both diffuse, low-amplitude signals and sharp, focal signals, we propose a simple but generic form of non-linear image processing that boosts the height of spatially distributed signals without changing the location of their maxima. This enables us to apply standard permutation testing to the height of the maxima of the resulting statistic image, while maintaining strong control over family-wise error. Critically, this avoids specifying a threshold on clusters, while sensitising the inference to a wide range of signal shapes; see Fig. 1 (right). Although the form of this threshold-free non-linear enhancement may seem ad hoc, we provide a series of heuristics in the appendices, which motivate its form and the algorithm's parameters.

This paper first presents the TFCE approach in detail, and provides some illustration of its characteristics. Detailed optimisation and validation is then presented, using a range of simulated signal types and noise levels, and careful ROC testing to compare the power of TFCE, cluster thresholding, voxel thresholding and a spatial wavelet approach. In the ROC testing we investigate both the control of family-wise error (comparing sensitivity when correcting for multiple comparisons), and also voxel-wise accuracy. TFCE appears to give sensible results, with generally better sensitivity and stability than the other methods. Appendices are included which further justify the TFCE method and specific parameter choices.

Section snippets

TFCE definition

The TFCE approach aims to enhance areas of signal that exhibit some spatial contiguity without relying on hard-threshold-based clustering. The image is passed through an algorithm which should enhance the intensity within cluster-like regions more than background (noise) regions. The output image is therefore not intrinsically clustered/thresholded, but the hope is that after TFCE enhancement, thresholding will better discriminate between noise and spatially-extended signal.

The TFCE algorithm

ROC-based evaluations using simulated datasets

In this section, simulated data comprising several test image shapes are used to compare various enhancement/thresholding methods against each other, with ROC evaluations giving objective combined measures of specificity and sensitivity.

ROC taking into account correction for multiple comparisons

Each of the methods was tested using a range of parameter settings, as described above. By careful investigation of the complete set of results for every method (including looking across all possible settings, as well as looking at summary scores, taking the arithmetic and geometric means of AUC values across all 3D test signals and all SNR values), we selected the “optimal” parameter values for each method. For “voxel”, the optimal smoothing was σ = 3 voxels. For “cluster”, we selected two

Real data example

We now give an illustration using real data. We used data published as part of a VBM-style analysis of early-onset schizophrenia (Douaud et al., 2007). Structural MR (T1-weighted) images from 25 adolescent-onset schizophrenic patients were compared with images from 25 healthy age- and gender-matched adolescents. Tools from FSL (Smith et al., 2004) were used to pre-process the data according to the “optimised-VBM” approach (Good et al., 2001). For each subject the Jacobian-modulated grey-matter

Discussion

In this paper we have presented a new, simple, approach for defining a cluster-like voxel-wise statistic in a way that we feel is more natural and stable than the commonly-used approach of an initial cluster-forming hard thresholding. TFCE enhances cluster-like features in a statistical image without having to define clusters as binary units. Through the use of permutation testing it is straightforward to control the FPR of the TFCE output image, including controlling for multiple comparisons

Acknowledgments

We are very grateful to the UK EPSRC for funding, to Matthew Webster for software coding, to Karla Miller for providing the high-resolution FMRI data, to Dimitri van de Ville for his help with WSPM, to Gwenäelle Douaud and Tony James for providing the schizophrenia data, and to Mark Woolrich, Adrian Groves, Karla Miller and Gwenäelle Douaud for useful discussions.

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