Embedding grayscale halftone pictures in QR Codes using Correction Trees

Barcodes like QR Codes have made that encoded messages have entered our everyday life, what suggests to attach them a second layer of information: directly available to human receiver for informational or marketing purposes. We will discuss a general problem of using codes with chosen statistical constrains, for example reproducing given grayscale picture using halftone technique. If both sender and receiver know these constrains, the optimal capacity can be easily approached by entropy coder. The problem is that this time only the sender knows them we will refer to these scenarios as constrained coding. Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as if both sides would know the constrains. We will analyze Correction Trees to approach analogous capacity in the general case use weaker: statistical constrains, what allows to apply them to all bits. Finding satisfying coding is similar to finding the proper correction in error correction problem, but instead of single ensured possibility, there are now statistically expected some. While in standard steganography we hide information in the least important bits, this time we create codes resembling given picture hide information in the freedom of realizing grayness by black and white pixels using halftone technique. We will also discuss combining with error correction and application to rate distortion problem.