. analysis of
[he wavelet
| transform-
rht complex
1,2, 3) (35)
rorof D. .
D;
D;
f an image
et analysis
4 SIMILARITY DISTANCE AND CONTINUOUS
MATCHING
Using the complex conjugate wavelet analysis, we can define
the similarity distance S;((z, y), (z', y')) for any pair of image
points on the reference image (e.g. left) f(z,y) and the
matched image (e.g. right) f'(z',y'). We shall use z' to
denote any thing on the matched image corresponding to z
on the reference image.
4.1 Implicit Feature Vectors
On a given j-th level of the wavelet pyramid, for a given po-
sition (z, y), we have 8 coefficients of complex conjugate
wavelet — — 3 analysis
(4;(z, 9), Aj(z, y), Dj p(x, 9), D,,p(2,9), p = 1,2,3).
Note that A, and A; are approximation components whose
information are further decomposed onto the next higher level
J+1. In order to match two images on the j-th level, the sim-
ilarity distance can be defined using only 6 differential com-
ponents D; »(z, y), Djs (z, y), p = 1,2,3. For image match-
ing to be invariant to local image intensity, a normalization
proved, with real image data, to be adequate, which leads to
the implicit feature vector B;(z, y)for each position (z, y)
: .(Dsa(m v) Djs(z,v) Dis(m,v)
Be) = (Gl Tse) TA, Gl
D;a(,9) Dia(z,9) Dis(a 4), (36)
|A;(æ,9)| |A;(æ,9)| |A;(=,9)|
where |.| denotes the module of a complex number. The
components B;p,p = 1,2,...,6 are also called subbands of
wavelet analysis.
4.2 Standard Similarity Distance
In the standard case where the relative rotation angle y is
small enough, a similarity distance S;((z, y), (z', y')) can be
defined as
Sio 3) (5^9) m M Sisen ^v? (37)
where Sj ,'s are the subband similarity distances, defined by
Sj, ((z, y), (z^ y')) = |B;,p(z,y) — Bj (s; y (38)
In a top-down hierarchical matching scheme, for an image
point (z — k, y — 1), (k,l) € Z?, on the reference image, we
may know its approximate correspondence (z^ zz k', y' zz I")
on the matched image. The precise correspondence may be
somewhere (z^, y") around (Xk', l^):
fk) — f'(k' - ul! v) (39)
where (u, v) € R?, denoting the differences
um! — k, v=y = (40)
Note (k,l) and (K', l^) take integer coordinates in the down-
sampled image on the level j. With this approximation, the
subband similarity distance is reformed to
Sip ((k 1), (K+ u,l' + v)) =
|Bp(k, 1) - Bi, (i ul v) — (41)
The best matching point in the simplest sense corresponds to
min Sj ((k, D), (k' -- u, l' -- v)) (42)
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
4.3 Continuous Interpolation for Fine Matching
In order to minimize the similarity distance S; ((k, 1), (k' 4
u, l' -F v)) in (42), we need to interpolate B7 ,(k' -- u, I 4- v).
Ideally, we should use four integer-indexed positions surround-
ing (k' -- u, l' + v), which leads to complicated interpolation
and minimization procedures. If we limit |u|, |v] € 0.5, we
can use single nearest neighbor (k’,1’) with a phase shift rel-
ative to the spatial shift (u, v),
B; ,(k' Fu, l 4 v) = BL 2 (K', ye? Biren) C (43)
where £2; denotes the pair of the modulation frequencies (in
X- and y-directions) for p-th subband D; , on the j-th level,
given by (forp=1,2,--.,6)
(95,05), (v5, 5), (v5, w;), (795,5), (75,95), (75 w;)
where w;, ©; are given inductively by (32) - (34).
It can be shown that with the continuous interpolation of
(43), the similarity distance can be written as
Sitfk, D, (k ud + v)) = si(u — uo)? + so(v — vo)”
+s2(u — vo)(v — vo) + 444)
where the coefficients s1,s2,53,54,%o0,vo can be com-
puted directly from given data B; (k,l), B; (k',I'), p =
1,2,...,6, and (w;,®;). (uo,v0) is the minimum point of
the similarity distance surface S;. s1, s2, s3 are the curva-
ture (second derivatives) along z-, y-, and diagonal directions
respectively, characterizing the uncertainty of the estimate
(wo, vo). sa is the minimum value of the similarity distance
S;-
4.4 Local Parallax Continuity and Generic Pattern
Matching
For the robustness of image matching, local parallax conti-
nuity should be taken into account when matching two given
positions (k,!) and (k’,1"). To maintain a compromise be-
tween fine locality of matching and robustness of matching,
we could define a new similarity distance P;((k, 1), (k', l)) in
the sense of generic pattern matching
P;((k, l), (k^, I") =
> mins;((&+r,1 + c),(k+r+ul+e+w)) | (45)
(no)
(r,c) € [(0,0), (—0.5, —0.5), (—0.5, .5), (0.5, —0.5), (0.5,0.5)]
where (u,v) is a fine-tuning shift vector, which is varia-
tional for each (r,c) pair. Note that (k + r,! + c) with
r,c = 0.5 or — 0.5 correspond to diagonal positions which
can also be computed with rigorous bottom-up wavelet trans-
form. Similarly, a normal position (k,l) may be matched to
a normal (k', i") or a diagonal position (k' + 7, l' + c).
5 SPIRAL AND HIERARCHICAL PARALLAX
PROPAGATION
5.1 Spiral Parallax Propagation
Without loss of generality, let us consider a stereo pair of
square images with size 2" x 2". With minimum overlap-
ping of 60%, the central area (e.g. 2 x 2 on the level of
8 x 8) on a chosen highest level of each image should have
correspondence on the other image. Image matching thus
can start with an exhaustive search for the best matching of
