3.2.8.1.2.2 Reduce-Extrapolate Method

The Original Method (section 3.2.8.1.2.1) for dealing with boundaries when encoding tiles introduces tile artifacts. The result is that users can perceive where the tile boundaries are in a decoded image. The Reduce-Extrapolate method removes this artifact.

The first pass for a given direction (horizontal or vertical) takes an input of 64 coefficients and produces 33 low-frequency results and 31 high-frequency results.

A 65th input coefficient is introduced by extrapolating from the last two input coefficients. Note that the subscripts used in the equations that follow are 1-based (in contrast to the equations in [MS-RDPRFX] section 3.1.8.1.4, which are 0–based). It is possible for the extrapolated 65th coefficient to lie outside of the normal pixel range. Furthermore, extrapolation is only required for the first level.

  Bands resulting from the Reduce-Extrapolate DWT Method        

The first-pass DWT is performed on the 65 coefficients, mirroring around the first and the sixty-fifth boundary elements. As a result, 33 low-frequency and 32 high-frequency results are obtained. The final frequency result is zero and is dropped.

 

     Bands resulting from the Reduce-Extrapolate DWT Method

The second-pass DWT takes the 33 low-frequency results from the first pass and performs a DWT with normal mirroring, producing in turn 17 low-frequency elements and 16 high-frequency elements.

Finally, the third-pass DWT takes the 17 low-frequency results and produces (using the same techniques as the previous pass) 9 low-frequency elements and 8 high-frequency elements.

The resulting bands and the sizes are illustrated in the following figure.

Bands resulting from the Reduce-Extrapolate DWT Method

Figure 3: Bands resulting from the Reduce-Extrapolate DWT Method