1-1. What is a generalized seamless texture?

   A seamless texture is invisible its boundary when they are tiled.  However ordinary seamless textures need a direction arranging,  therefore each texture pierce can be perceived easily by seeing regular pattern repeats .  We have developed generalized seamless texture which can be tiled independent of its direction and boundary is invisible.

1-2. A technical point
 
  The most difficult problem in synthesizing generalized seamless textures is keeping texture statistical uniformities between border and inside.  A synthesized texture without consideration of this problem usually have an peculiar pattern like a picture frame.  We found out the cause is its topology. A topology of ordinary seamless textures is torus, however it of generalized seamless textures is not torus, even not manifold, therefore vertexes and edge lines of them are singular in a differential topology sense.  We have proposed a few of synthesizing technique to avoid or reduce this affection. (In principle, no perfect solution exists.)
 
 

                A sample of generalized seamless textures (3*4 square pieces are tiled with random directions)

2-1. A problem of finit length digitalized data

  All analog-to-digital recorders translate an analog signal into finite digital data, although an analog signal has infinity information quantity. The loss of information quantity in this process cause to be quality down.  In case of CD (Compact Disc), a play backed signal includes quantization error of -96 dB amplitude because of 16 bit quantization. Our purpose is estimating an original analog signal from a digitalized music data by rejecting quantization error. The subject at present is getting a signal the quality of which is equivalent to a 20 bit quantization data.  

Attention:
  Restoring a data of 20 bit accuracy from a 16 bit data is different from restoring 20 bit information quantity from 16 bit information quantity. There is no way to do the latter.  Even the former can succeed, net information quantity isn't increase.

 2-2. Frequency Vector Smoothing Method (FVSM)   Now researching!
 
  We have proposed two assumptions for an estimation.
1. Both frequency amplitude and phase of musical signals vary linearly in an enough short time interval (< 10 mSec). This assumption is expected to hold good in most of time sequence except transient interval (e.g. attacking, tonguing ...).
2. Statistical character of quantization error on CD is equivalent to it of a white noise of -96 dB. This assumption hold good under using usual musical signals except perfect regular signals like a sine wave.
  We have developed Frequency Vector Smoothing Method (FVSM) by using these assumptions. FVSM has effects as a kind of correlating filter and also Wiener Filter implicitly.  An experiment by using artificial stationary waves shows quantization error reducing at -6 dB.  This mean that FVSM can restore a data of 17 bit accuracy from a 16 bit data.