Barycenter of Images according to Cuturi & Doucet (2014)

Using entropic regularization for Wasserstein barycenter computation, imagebary14C finds a barycentric image \(X^*\) given multiple images \(X_1,X_2,\ldots,X_N\). Please note the followings; (1) we only take a matrix as an image so please make it grayscale if not, (2) all images should be of same size - no resizing is performed.

Usage

imagebary14C(images, p = 2, weights = NULL, lambda = NULL, ...)

Arguments

images

a length-\(N\) list of same-size image matrices of size \((m\times n)\).

p

an exponent for the order of the distance (default: 2).

weights

a weight of each image; if NULL (default), uniform weight is set. Otherwise, it should be a length-\(N\) vector of nonnegative weights.

lambda

a regularization parameter; if NULL (default), a paper's suggestion would be taken, or it should be a strictly positive real number.

...

extra parameters including

abstol: stopping criterion for iterations (default: 1e-8).
init.image: an initial weight image (default: uniform weight).
maxiter: maximum number of iterations (default: 496).
nthread: compatibility control; only 1 is supported.
print.progress: a logical to show final solver diagnostics (default: FALSE).

Value

A matrix of image probability masses with a diagnostics attribute; return_result = TRUE returns a rich fit and image.

Details

Since version 0.2.0 the historical names call the shared objective-checked mirror-descent solver with log-domain Sinkhorn gradients, as documented in fbary14C. The entropy coefficient lambda multiplies sum(plan * (log(plan) - 1)). Pixel columns run from x=0 to x=1 and rows from y=1 to y=0; a singleton axis is fixed at its first coordinate. Pixel intensities are probability masses after normalization. Additional controls from fbary14C are accepted, using init.image instead of init.vec.

References

Cuturi M, Doucet A (2014-06-22/2014-06-24). “Fast Computation of Wasserstein Barycenters.” In Xing EP, Jebara T (eds.), Proceedings of the 31st International Conference on Machine Learning, volume 32 of Proceedings of Machine Learning Research, 685–693.

Examples

if (FALSE) { # \dontrun{
#----------------------------------------------------------------------
#                       MNIST Data with Digit 3
#
# EXAMPLE 1 : Very Small  Example for CRAN; just showing how to use it!
# EXAMPLE 2 : Medium-size Example for Evolution of Output
#----------------------------------------------------------------------
# EXAMPLE 1
data(digit3)
datsmall = digit3[1:2]
outsmall = imagebary14C(datsmall, maxiter=3)

# EXAMPLE 2 : Barycenter of 100 Images
# RANDOMLY SELECT THE IMAGES
data(digit3)
dat2 = digit3[sample(1:2000, 100)]  # select 100 images

# RUN SEQUENTIALLY
run10 = imagebary14C(dat2, maxiter=10)                   # first 10 iterations
run20 = imagebary14C(dat2, maxiter=10, init.image=run10) # run 40 more
run50 = imagebary14C(dat2, maxiter=30, init.image=run20) # run 50 more

# VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,3), pty="s")
image(dat2[[sample(100,1)]], axes=FALSE, main="a random image")
image(dat2[[sample(100,1)]], axes=FALSE, main="a random image")
image(dat2[[sample(100,1)]], axes=FALSE, main="a random image")
image(run10, axes=FALSE, main="barycenter after 10 iter")
image(run20, axes=FALSE, main="barycenter after 20 iter")
image(run50, axes=FALSE, main="barycenter after 50 iter")
par(opar)
} # }