Robust TV: Huber and Tukey Data Terms

Robust losses tolerate outliers better than the standard squared error. In fMRI, spikes or motion corruption can bias classical TV. The Huber loss behaves quadratically for small residuals and linearly for large ones; Tukey’s biweight further down‑weights extreme outliers.

stv_robust4d() implements robust data terms with space–time TV priors. Thresholds are derived from a noise estimate if not set explicitly.

Parameters in practice

lambda_s and lambda_t are the TV weights as in standard TV. Select loss = "huber" or loss = "tukey". The Huber threshold delta and Tukey cutoff cthresh determine the transition from inlier to outlier behavior; leaving them NULL estimates them from the data. The alpha parameter tunes the Tukey shape.

Temporal weights can reduce smoothing across high‑motion transitions: provide per‑frame motion parameters via motion_params, or pass explicit temporal_weights.

Example

d4 <- c(8, 8, 8, 10)
base <- array(100, dim = d4) + array(rnorm(prod(d4), sd = 5), dim = d4)
base[4,4,4,5] <- 220  # inject an outlier

rob_huber <- tv_robust4d(base, loss = "huber", iters = 20L)
rob_tukey <- tv_robust4d(base, loss = "tukey", iters = 20L)

c(var_huber = var(as.vector(rob_huber)),
  var_tukey = var(as.vector(rob_tukey)))
#> var_huber var_tukey 
#>  28.89872  28.90254

# visualize the frame containing the outlier, mid-slice
zmid <- ceiling(d4[3]/2); t_spike <- 5
viz <- rbind(
  slice_df4d(base, zmid, t_spike, "noisy+outlier"),
  slice_df4d(rob_huber, zmid, t_spike, "Huber"),
  slice_df4d(rob_tukey, zmid, t_spike, "Tukey")
)
ggplot(viz, aes(x, y, fill = val)) + geom_raster() + coord_fixed() + scale_y_reverse() +
  scale_fill_gradient(low = "black", high = "white") + facet_wrap(~method) + theme_minimal(base_size = 10) +
  labs(title = "Robust TV on an outlier frame (mid-slice)", x = NULL, y = NULL, fill = "intensity")

The robust losses reduce the influence of isolated spikes relative to standard TV. When artifacts correlate with motion, supplying motion_params improves temporal selectivity.

fmrismooth authors

2025-08-28

Parameters in practice

Example