Variable-coefficient generalizations of integrable nonlinear equations are one of exciting subjects in the study of mechanical, physical, mathematical and engineering sciences. In this paper, we present a KdV family (namely, KdV, modified KdV, Calogero-Degasperis-Fokas and Harry-Dym equations) with variable coefficients in (2 + 1) dimensions.

In this paper, a coefficient-parameter reduction method is proposed for invertible deinterlacing with variable coefficients. Invertible deinterlacing, which the authors have developed before, can be used as a preprocess of frame-based motion picture codec, such as Motion-JPEG2000 (MJP2), for interlaced videos. When the conventional field-interleaving is used instead, comb-tooth artifacts appear around edges of moving objects. On the other hand, the invertible deinterlacing technique allows us to suppress the comb-tooth artifacts and also to recover an original picture on demand. As previous works, the authors have developed a variable coefficient scheme with a motion detection filter, which realizes adaptability to local characteristics of given pictures. When applying this deinterlacing technique to an image codec, it is required to send coefficient parameters to receivers for original picture recovery. This work proposes a parameter decimation technique and shows that this reduction approach can be achieved without significant loss of comb-tooth suppression capability and improves the quality at high bit-rate decoding.