International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
In (Anandan, 1989) SSD-function is sequentially optimized by 
the Laplacian pyramid. Minimum is found for all levels of the 
pyramid to begin with the highest level (the smallest image) and 
dropping to the lowest level (the whole image). Speed vector is 
being obtained more accurate on each level. In (Singh, 1992) 
minimum of SSD-function is found through iteration process. 
But correlation approach is not robust too because it strongly de- 
pends on invariability of scene brightness. In (Heeger, 1988) fre- 
quency approach is proposed. This approach is based on “power” 
function, evaluated as the Gabor filter (Gabor, 1946) with fre- 
quencies Lz, Ly, w: 
  
R(u,v) = nd —4n?o2070, (uLz vL, + w) } 
(uozok)? + (voyok)? + (020y)? 
where 95, 0, 04, — standard Gabor filter derivatives. 
Speed vector (u, v)? is found during minimization of function 
12 
it M JF, (u,v) 
flu, v) = 2 mi ne ue) 
with respect to u and v, where m; — measured power value, R; — 
predicted power value, m; and R; are average power values. 
3 REGRESSION PSEUDOSPECTRUMS 
In this section we introduce the notion of multiple-regression 
pseudospectrums. 
Let again /(k) be an input image pixel matrix with width w and 
height h on frame number k, I(k) € R"*^,. It is assumed that 
I(%) is a grayscale image, so 0 < I(k),, «€ 255 Vr = 
l...w, y = 1...h Let us call Mn(k) an regression accu- 
mulator of n frames with parameter à, calculated on frame k. It 
will be a matrix Mn (k) € R**"(Box et al., 1994): 
M,(k + 1) = aM,(k) + (1 — a)I(k). (1) 
You can calculate the accumulator value M,(k) on frame k by 
adding each older member in series (1): 
e 
E 
Ma(k) — (1-0) M. a^ 7I. Q) 
iz0 
ll 
Let us assume that [(k) is an element of the image matrix I (k) 
and mn(k) is an element of the accumulator matrix M,, (k) with 
the same coordinates, as [(k). Let us suppose that on an initially 
zero input of accumulator (2) since some moment ko (without 
loss of generality, let ko — 0), during enough long time some 
signal with intensity / is being given: 
k—1 
ma(K) 7 I(1— 0) 7 a*1 zi - o^). (3) 
i=0 
Now it's quite simple to find such o, so that m,, (k) would surely 
exceed [3 share of signal [ after n frames: 
Mn{n) = I(1 — a”) = Bl. 
Hence 
Un = Yi = B. (4) 
560 
Thus, o is such time averaging parameter, at which the accu- 
mulator sum will be equal to m,(n) = SI through n frames. At 
the same time n here can be called 5 memory length or, simply, 
length of accumulator memory with the corresponding averaging 
parameter o, — V1- f. 
Given o; can be found as (4), the whole accumulator sum in one 
pixel at variable frame k can be found as 
ma(k) 2 (1 — o5) 2 1(1— (1 — 8)//^). (5) 
The m,(k) graphs for different an,n = 4, 8, 16, 32 values are 
shown in Figure 1, supposed 4 = 0.5,/ = 100, ko = 10. 
  
HS 
  
  
  
  
  
  
  
3 
Lo 
3 » signal! 
I 
aeu 
dei 
quia 
5i i 
ü m 20 
x 
frame number 
Figure 1: Accumulated pixel mn (k) for different an. 
Thus, time averaging parameter œn, defined in (4), is in fact the 
satiety parameter of the filter response function. It allows to judge 
after which time (in frames) n accumulated sum will be equal to 
Bl. 
According to (5), œn possesses the multiplicity property: 
Un = One: (6) 
Indeed, 0? ; — ( nl B) = V1- 8 = an. Let us call 
Dn,s(k) = mn(k) — Mn-s(k) 
a difference between the responses of accumulators with multi- 
ple smoothing parameter n and n - s. By (6) and assuming that 
some signal with intensity / is being given from time ko = 0, this 
difference will possess a very interesting property: 
  
Dy s(n:5s) 2 ma(n- s) — mas(n-5) — (7) 
-i(1-a-5*) -i(1-a- 9*9) = 
-i0-9(-0-873)-11-9)Y-g. 
Consider the behaviour of derivatives D, (k) function. Let s — 
2 and 8 — 0.5. Then, according to (7), difference between accu- 
mulator with memory of 2n frames and accumulator with mem- 
ory of n frames will be equal to 
Da a(2n) z: 0.951. (8) 
Figure 2 shows differences D,, s between accumulators with vari- 
able n and s = 2,1 = 100. 
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