We Claim:
1. An image encoding device comprising:
an image compressor that carries out a transformation process on a difference image between an inputted image and a prediction image, quantizes transform coefficients of the difference image, and outputs the quantized transform coefficients as compressed data;
a filter that carries out a filtering process on a decoded image which is a result of addition of the difference image decoded from the compressed data and the prediction image; and
a variable length encoder that variable-length-encodes the compressed data, an adaptation parameter set and a filter parameter used when the filtering process is carried out by the filter and generates a coded bitstream into which coded data of the compressed data and coded data of the filter parameter are multiplexed,
wherein the filter determines a classification method of a class on each coding block having a largest size, carries out a classification on each pixel within each coding block having the largest size by using the classification method, calculates an offset value for each class for each coding block having the largest size, and carries out a pixel adaptive offset process by which the offset value is added to a pixel value of a pixel belonging to a corresponding class, and
the variable length encoder variable-length-encodes a parameter associated with a quantization matrix for generating a quantization matrix used when the transform coefficients are quantized by the image compressor and an index indicating the classification method of a class on each coding block having the largest size, the classification method being determined by the filter, and variable-length-encodes a parameter of the offset value for each class on a basis of a binarization process using a truncated unary code, and

the adaptation parameter set includes a flag showing that the parameter associated with the quantization matrix exists and the parameter associated with the quantization matrix.
2. An image decoding device comprising:
a variable length decoder that variable-length-decodes compressed data, a filter parameter and an adaptation parameter set from coded data multiplexed into a coded bitstream;
a difference image generator that inverse-quantizes transform coefficients of a block obtained from the compressed data by using a parameter associated with a quantization matrix included in the adaptation parameter set which is variable-length-decoded by the variable length decoder and inverse-transforms the inverse-quantized transform coefficients to generate a difference image;
a decoded image generator that adds the difference image and a prediction image to generate a decoded image; and
a filter that carries out a filtering process on the decoded image decoded from the compressed data by using the filter parameter,
wherein the variable length decoder variable-length-decodes, as a filter parameter, an index indicating a classification method of a class on each coding block having a largest size from the coded data, and variable-length-decodes a parameter of an offset value for each class on which a binarization process using a truncated unary code is performed, and
the filter specifies the classification method of a class on each coding block having the largest size by using the index, carries out a classification on each pixel by using the classification method, and carries out a pixel adaptive offset process by which the offset value is added to a pixel value of a pixel belonging to a corresponding class, and
the adaptation parameter set includes a flag showing that the parameter associated with the quantization matrix exists and the parameter associated with the quantization matrix.