Space–Time Total Variation Denoising

fmrismooth authors

2025-08-28

Total variation (TV) denoising assumes the underlying image is piecewise smooth and penalizes the sum of absolute gradients. In space–time fMRI, this preserves edges while reducing high‑frequency noise across both spatial and temporal axes.

fmrismooth provides a 4D ROF‑type TV model solved by a primal–dual algorithm.

Parameters and their effects

lambda_s controls spatial smoothing; higher values produce flatter regions between edges. lambda_t controls temporal smoothing; increasing it enforces consistency across frames. The algorithm step sizes tau, sigma, and relaxation theta are set to robust defaults but can be tuned.

Example

d4 <- c(10, 10, 10, 12)
base <- array(100, dim = d4)
base[6:10, , , ] <- base[6:10, , , ] + 15
noisy <- base + array(rnorm(prod(d4), sd = 5), dim = d4)

tv_moderate <- tv_denoise4d(noisy, lambda_s = 0.6, lambda_t = 0.2, iters = 20L)
tv_strong   <- tv_denoise4d(noisy, lambda_s = 0.9, lambda_t = 0.35, iters = 20L)

c(var_noisy   = var(as.vector(noisy)),
  var_mod     = var(as.vector(tv_moderate)),
  var_strong  = var(as.vector(tv_strong)))
#>  var_noisy    var_mod var_strong 
#>   81.30328   64.57129   59.07842

zmid <- ceiling(d4[3]/2); tmid <- ceiling(d4[4]/2)
viz <- rbind(
  slice_df4d(noisy, zmid, tmid, "noisy"),
  slice_df4d(tv_moderate, zmid, tmid, "TV moderate"),
  slice_df4d(tv_strong, zmid, tmid, "TV strong")
)
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 = "TV denoising: mid-slice/frame", x = NULL, y = NULL, fill = "intensity")

TV excels when you want aggressive noise reduction with crisp boundaries. If small features are oversmoothed, reduce lambda_s and/or lambda_t, or try a bilateral or guided filter.