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Reading the report here, they use 'Log-Domain Integration'.

It is said to be a type of 'Log-Domain' Filter, but googling it, there is just a bunch of reports/articles on it.

Can anyone explain what Log-Domain filtering is and why we might use it over?

For very low supply voltages (1V or even lower) we can use the principle of analog signal processing in the "log domain". Using this principle, the exponential current-voltage characteristic of bipolar transistors (for very small currents) or of MOSFETs (sub-threshold region) is exploited.

The principle is based on a kind of "companding": The input signal is a current, which produces a "compressed" voltage in the log-domain - and after signal processing in this domain (amplification, intergation, filtering) the voltage signal is expanded and transferred back again into a current using the exponential characteristic of a transistor (inverse operation if compared with the input expansion).

In all cases, log-domain signal processing is realized in form of integrated circuits only. And as the main advantage very low supply voltages are allowed.

Literature:

Here is a good introductory text:

http://bioelectronics.tudelft.nl/~wout/documents/iscas20022.pdf

According to the first google result 'what is log domain filtering?':

Synthesis of Log-Domain Filters from First-Order Building Blocks

First part of 1. Introduction:

Definition:

Log-domain filters comprise a subclass of circuits
having externally linear transfer functions but
internally nonlinear components [1,2]. As the name
implies, log-domain filters are specifically those
circuits whose internal state is a logarithmic function
of the input and output. The circuit design exploits this
particular nonlinearity directly rather than attempting
linearization around an operating point. The result is
that log-domain filters have large-signal linearity: the
transfer function describes the overall behavior of the
system. 

Usage:

The equations governing the internal nonlinearity
of the system are generally tractable, leading
to complete solutions which do not require separate
DC and transient analyses.