X ANALYSIS SYNTHESIS
2.8. Properties of Wavelet Filters
2.8.1 Regularity and Vanishing moments
Beylkin, Coifm an, an d Rokhlin [45] u sed com pactly su p p o rte d
o rth o n o rm al w avelets to com press large m atrices, i.e. to reduce th em to a sparse
form . O ne of the things th at m akes their m eth o d w o rk is the n u m b e r of
van ish in g m om ents. Suppose w e w a n t to decom pose a function / into
w avelets. W e com pute all the w avelet coefficients ^^id to com press all
the inform ation, w e discard all the coefficients sm aller th a n som e th re sh o ld e.
precision to som e coefficients th an to others, b y m eans of a q u an tizatio n rule).
A fter thresholding, w e w ill only retain fine-scale w avelet coefficients near
singularities of / or its derivatives. The effect w ill be all the m ore p ro n o u n ced
if the n u m b er N of vanishing m om ents of \{/ is large. N ote th a t the regularity
of \{/ does n o t play a role at all in this argum ent, it seem s th a t for com pression
of m atrices-type applications the n u m b er of van ish in g m om ents is far m ore
im p o rta n t th a n the regularity of y/ .
For o th er applications, regularity m ay be m ore relevant. Suppose w e w a n t to
com press the inform ation in an image. A t least som e reg u larity m ig h t be
required. Some initial experim ents rep o rted b y A ntonini et.al. [46] seem to
confirm this, b u t m ore experim ents are req u ired for a convincing answ er.
The idea of generating regular functions from a rep eated in terp o latio n schem e
is n o t new . It appears in com puter-aided geom etric design, an d the dependence
of 0 (0 on Fo(z) w as also observed by Burt an d A delson in the context of
p y ra m id transform s [34]. These w orks w ere perfo rm ed in d ep en d en tly of
w avelets w h ere regularity does n o t d ep en d on the perfect reconstruction
p ro p e rty of filterbanks. But w h en intro d u cin g com pactly su p p o rte d w avelets,
D aubechies stated new problem s: u n d e r w hich (necessary a n d sufficient)
conditions o n Fq(z) do w e have convergence of c[ to 0 ( 0 a n d reg u larity of
the lim it function?
These questions are m otivated by the m athem atics b eh in d regularity, w h ere the
reasoning com es from w avelet analysis of continuous signals. But this
m athem atical ap p ro ach seem s inad eq u ate for digital signal processing
n u m b er of times. In w avelet based im age or speech com pression schem es,
i seldom exceeds 5, w hereas regularity is m athem atically defined w h e n / —> .
H ow ever, w e know from M allat's w o rk th at lim it functions still u n d e rla y a
discrete filter bank, even w h en it is n o t iterated indefinitely [47]. Therefore, w e
w o u ld like to u n d e rsta n d regularity in term s of the discrete filter b an k im pulse
responses c[ an d b[ .
R egularity does im pose som e "sm oothness" on c[ an d b[ . There are intuitive
arg u m en ts in favour of this for coding applications.
• D uring analysis: Suppose th at a sm ooth p o rtio n of the in p u t is analysed by
"n o n reg u lar" filters, w hose im pulse responses rap id ly cause discontinuities
as i increases. Then, these artificial discontinuities, n o t d u e to the signal
itself, ap p ear in the w avelet transform coefficients. In other w o rd s, reg u larity
w o u ld lead to a "better" representation of the signal b y these coefficients.
• D uring synthesis: Suppose th at an error, (a quan tizatio n error), is m ad e in
one coefficient corresponding to som e decom position level i. In the
reconstructed signal, this results in a p e rtu rb atio n th a t is p ro p o rtio n al to the
equivalent im pulse response corresponding to this level. In applications such
as im age com pression, a p ertu rb atio n p resen tin g discontinuities is likely to
"strike the eye" m ore th an a sm ooth one. W hile in applications such as audio
com pression it is likely to "irritate the ear" by p ro d u cin g glitches. Also, its
am p litu d e increases for hig h com pression rates, w h en transform coefficients
are coarsely quantized. Therefore, it is n a tu ra l to require th a t this
Rioul [43] pro v id es an exam ple of im age coding application, in w hich the effect
of reg u larity on com pression perform ance is m easured, u sing o rthogonal
w avelets. From this perform ance com parison for different filter lengths it seem s
th a t the coding perform ance globally increases w ith filter length, w hich also
increases th e regularity. H ow ever, an asym ptote is quickly attained. A bove L =
10 or 12, for w hich the regularity order does n o t exceed 2, perform ance does n o t
im p ro v e m uch.
Results obtained for a sim ple com pression scheme u sin g vario u s coding criteria,
optim ised ra te /d isto rtio n , an d a num ber of p a ra u n ita ry filters w ith balanced
reg u larity an d frequency selectivity, show th at regularity m ay be relev an t for
still im age com pression, at least for short filters (L<10), for w hich the
reg u larity o rd er is relatively small. Using m ore reg u lar filters is p robably
useless, as the com pression perform ance is n ot im proving for longer filters.
Silva a n d G hanbari [48] using biorthogonal w avelets, m easu re also th e effect of
reg u larity an d the nu m b er of vanishing m om ents on th e com pression
p erform ance for im age coding applications. It has b een verified th a t as long as
the reg u larity of th e analysis w avelet is larger th an 1, the increase in reg u larity
does n o t enhance the "coding gain" [40] of a w avelet transform . O n th e other
h an d , if the reg u larity is less th an 1, then the low er is the reg u larity the sm aller
becom es the "coding gain".
O r in oth er w ords, their results show that as long as the "coding g ain" of a
w avelet transform is reasonably high, the influence of b o th reg u larity an d
for w avelet transform s w ith "coding gains" < 1, the "coding gain" decreases
w h e n th e reg u larity decreases.
Sinha an d Tewfik [49] suggest th at regularity can play a role in th e coding of
au d io signals. In fact they claim th at longer sequences yield b etter results: This
conclusion in n o t su rp risin g since the longer sequences (thus m ore regular
w av elet filters) correspond to w avelet filter banks w ith sh arp er tran sitio n
b an d w id th s, i.e. to a b etter separation of frequency inform ation. For a typical
au d io signal the average b it rate for quantizing the w avelet coefficients w as 2.1
b its/s a m p le w h en an optim al 4 coefficients w avelet w as used. The b it rate w en t
d o w n to 1.75 b its/s a m p le w h en an optim al 20 coefficients w avelet w as used,
a n d to 0.8 b its/s a m p le w h en an optim al 40 coefficients w as used.
H ow ever, Barnell an d Richardson [50] have different opinions: A nalysis-
synthesis system s based on filter banks, iteratively satisfy a set of constraints on
the filters an d the reconstruction properties of the analysis-synthesis system . To
m ake the system into a DWT, in general an extra set of constraints (m ainly
associated w ith regularity) m ust be satisfied. The p rim ary function of the
reg u larity constraint in w avelets is to guarantee th a t the to tal octave b an d
decom position w ill converge, in the limit, to a sm ooth function. In practice,
ho w ev er, DTWTs m u st be realised w ith a finite n u m b er of b an d s, so the
reg u larity constraint generally results in subband system s w hose perform ance
is m easurably w orse in term s of filter perform ance (passband, sto p b an d ,
tra n sitio n band), reconstruction error, delay, com pactness (length),
orthogonality, com putational com plexity, etc. as com pared to system s w hich do
2.8.2. Symmetry
If the restriction th at (j) be real is lifted, th en sym m etry is possible, even
if 0 is com pactly su p p o rted . The need for sym m etry is unclear, since it h as been
d em o n strated th at for som e applications it does n o t really m atter at all. The
num erical analysis in Beylkin, Coifm an, an d R okhlin [45], for instance, w orks
very w ell w ith asym m etric w avelets. For oth er applications, the asy m m etry can
be a nuisance. In im age coding, for exam ple, q u an tizatio n erro rs w ill often be
m ost p ro m in en t aro u n d edges in the images. It is a p ro p e rty of o u r visual
system th a t w e are m ore tolerant of sym m etric errors th an asym m etric ones. In
oth er w o rd s, less asym m etry w o uld result in greater com pressibility for the
sam e p ercep tu al error. (N ote also th at "perceptually" sm all or large erro rs are
difficult to quantify m athem atically. The n o rm m ost often u sed to m easu re the
"distance" is the /^-norm , b u t it is m ore because this is the easiest n o rm to
h an d le th a n for any other reason. All experts agree, th a t the Z^-norm is n o t a
good candidate for a "perceptual" norm , b u t there does n o t seem to be an
arg u m en t on a better candidate). M oreover, sym m etric filters m ake it easier to
deal w ith the b o u n d aries of the im age, another reason w h y the su b b an d coding
engineering literature often sticks to sym m etry. W e can recover sym m etry if w e
2.9 Summary
Finally, large n u m b er of vanishing m om ents of y/ leads to m u ch m ore
""compression potential" in the regions w h ere / is reasonably sm ooth. It is n o t
clear how ever, w h eth er the high n u m b er of v anishing m om ents of y/ or the
reg u larity of y/ is the m ost im p o rtan t factor, it is possible th at th ey are b o th
im portant. H ow ever, sym m etry can be achieved by usin g Coiflets or
bio rth o g o n al w avelet bases. N ote also, th at for com putational p u rp o ses, linear