Tim Morrison wrote:http://en.wikipedia.org/wiki/Lossless_JPEG seems to give a pretty good description. A couple of quotes -
Lossless JPEG refers to a 1993 addition to JPEG standard by the Joint Photographic Experts Group to enable lossless compression. However, it might be used as an umbrella term to refer to all lossless compression schemes developed by the Joint Photographic Expert group. They include JPEG 2000 and JPEG-LS.
... and a description of the different encoding method used -
Lossless JPEG is actually a mode of operation of JPEG. This mode exists because the Discrete Cosine Transform (DCT) based form cannot guarantee that encoder input would exactly match decoder output since the Inverse DCT is not rigorously defined. Unlike the lossy mode which is based on the DCT, the lossless coding process employs a simple predictive coding model called differential pulse code modulation (DPCM).
yes that is lossless jpeg. that is not what is implemented when PSP X4 saves as 'lossless' jpeg.
the phrasing of the wikipedia article bight be clearer like this:
Lossless JPEG is actually a mode of operation of JPEG. This mode exists because the Discrete Cosine Transform (DCT) (used by the standard, lossy, jpeg encoder,) based form cannot guarantee that encoder input would exactly match decoder output since the Inverse DCT is not rigorously defined. Unlike the lossy mode which is based on the DCT, the lossless coding process employs a simple predictive coding model called differential pulse code modulation (DPCM).
the PSP X4 'lossless' jpeg is DCT based.
DCT incidentally is a non algebraic fourier transform technique where discrete data is approximated using cosine functions. The image below shows how different frequency, in this case sine waves are combined to produce something approximating a square wave.

but no matter how many you use it is never completely accurate because of the Gibbs phenomenon.