  We review a class of methods that can be collected under the name nonlinear
transform coding (NTC), which over the past few years have become competitive
with the best linear transform codecs for images, and have superseded them in
terms of rate--distortion performance under established perceptual quality
metrics such as MS-SSIM. We assess the empirical rate--distortion performance
of NTC with the help of simple example sources, for which the optimal
performance of a vector quantizer is easier to estimate than with natural data
sources. To this end, we introduce a novel variant of entropy-constrained
vector quantization. We provide an analysis of various forms of stochastic
optimization techniques for NTC models; review architectures of transforms
based on artificial neural networks, as well as learned entropy models; and
provide a direct comparison of a number of methods to parameterize the
rate--distortion trade-off of nonlinear transforms, introducing a simplified
one.
