BiExponential Transformation is a name given to a visualization technique used to make high-resolution compensated digital flow cytometry data more appealing to the eye. As a visualization technique, this does not modify your data - it's similar to distorting an image by bending/warping a mirror. According to the Herzenberg paper linked below, it's a requirement for anyone wishing to get the appropriate picture of this type of data to correct for the visual artefacts associated with negative-event fluorescence and picket fencing, both shown here.

After the data have been compensated using FlowJo, open a graph of a compensated population. From one or both of the axis labels, choose a compensated parameter. A small square button with a T will appear next to the axis label. Click this buttonto display options.

Here you can select between displaying linear or logarithmic display.
If the population has compensation applied, select Auto-calculate... to have FlowJo display extra negative decades to accomodate the data. If this is not selected, FlowJo will use the general preference you set for negative decades. NOTE: this enacts a change in display and not in your data. The compensated fluorescence values for each cell are EXACTLY THE SAME as before the transformation. The display is changed, but NOT the data. Statistics computed on a particular set of cells will be the same. The new display just makes it easier to identify correct population boundaries.
Apply to all (ForSc) - uses your choice of log or lin for the current parameter in each graph opened from the current workspace.
Apply to all compensated parameters - uses your choice of log or lin for all compensated parameters from the current workspace.
Remove Transforms - Displays the current axis without BiExponential Transformation.

Transformation Method - The display transformation process is based on a method developed by Dave Parks and Wayne Moore at Stanford University using a generalization of the hyperbolic sine function. The display functions approach true log for high data values and approach true linear around zero. This provides smooth, near-linear display of low and negative data values.

Notes - Digital Linear Data can also be transformed to provide a more interpretable view instead of the "picket fences" that occur at the low end of 5+ decade log scales. Click here for more information.

The transformation can be used on any compensated data--but FlowJo needs to be the one to compensate the data. Therefore, it can be done on any data where you collect the comp controls and create the compensation matrix, or, alternatively, on any data files which specify their own comp matrix (currently, only BD DiVa files do this correctly).

For a more detailed look at data transformations, refer to the papers below:

Interpreting flow cytometry data: a guide for the perplexed.
Herzenberg, Tung, Moore, Parks.

A New ‘‘Logicle’’ Display Method Avoids Deceptive Effects of Logarithmic Scaling for Low Signals and Compensated Data.
Parks, Roederer, Moore.