# Lair Of The Multimedia Guru

## 2005-11-27

### The snow wavelet

The default wavelet used by the snow codec in ffmpeg is a symmetric biorthogonal compact 9/7 wavelet with rational coefficients with properties very similar to the famous biorthoginal daubechies 9/7 wavelet, it was found by bruteforce search with the goal of having a simple lifting implemenattion and having many vanishing moments or almost vanishing moments, the daubechies wavelet has (4,4) our wavelet has (4,2) vanishing moments
The lifting coefficients are:

dau97 snow
-0.443506852 -15/32
-0.882911075 -4/5
0.052980118 1/16
1.586134342 3/2

The resulting wavelet and scaling functions look like:

One thing which is still annoying is the 4/5, while this can be approximated pretty well by a multiplication and a shift right but this tends to overflow so something like

```x += x+x;
x += x>>4;
x += x>>8;
x >>= 2;
```

is needed, this one only differs from 4/5 by a factor of ~0.00001, alternatively the 4/5 can be merged into the unquantization step so that

```\ | /|\ | /|\ | /|\ | /|\
\|/ | \|/ | \|/ | \|/ |
+  |  +  |  +  |  +  |   -15/32
/|\ | /|\ | /|\ | /|\ |
/ | \|/ | \|/ | \|/ | \|/
|  +  |  +  |  +  |  +   -4/5
\ | /|\ | /|\ | /|\ | /|\
\|/ | \|/ | \|/ | \|/ |
+  |  +  |  +  |  +  |   +1/16
/|\ | /|\ | /|\ | /|\ |
/ | \|/ | \|/ | \|/ | \|/
|  +  |  +  |  +  |  +   +3/2
is changed to
\ | /|\ | /|\ | /|\ | /|\
\|/ | \|/ | \|/ | \|/ |
+  |  +  |  +  |  +  |   -3/8
/|\ | /|\ | /|\ | /|\ |
/ | \|/ | \|/ | \|/ | \|/
(|  + (|  + (|  + (|  +   -1
\ + /|\ + /|\ + /|\ + /|\  +1/4
\|/ | \|/ | \|/ | \|/ |
+  |  +  |  +  |  +  |   +1/16
/|\ | /|\ | /|\ | /|\ |
/ | \|/ | \|/ | \|/ | \|/
|  +  |  +  |  +  |  +   +3/2
```

Now lets compare quality between the snow and daubechies 9/7 wavelets, this comparission has been done with JasPer with this change and tinypsnr (from ffmpeg) The used test image was the 256×256 lena grayscale image

Daubechies 9/7 wavelet Snows 9/7 wavelet
size PSNR
618 24.02
1283 27.32
3194 31.80
6503 36.82
13079 42.93
22945 47.14
size PSNR
622 24.03
1295 27.33
3212 31.82
6519 36.83
12981 42.87
22941 47.18
Filed under: Wavelets — Michael @ 22:09