valeri labunets - fast multiparametric wavelet transforms and packets for image processing
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Multiparametric Wavelet Transforms
E.Ostheimer2, V.G. Labunets1, D.E.Komarov1,
Yekaterinburg , AIST-2016
1Ural Federal University, pr. Mira, 19, Yekaterinburg, 620002, Russian Federation
2Capricat LLC 1340 S. Ocean Blvd., Suite 209 Pompano Beach 33062 Florida USA
WDT is characterized by two sets of coefficients
0 1 1
20 1 1
, ,...,, where L=2D
, ,...,n
L
L
h h hg g g
WDT
Structure of WDT
Relations between g- and h-coefficients:
0 1 1
20 1 1
, ,...,, ,...,n
L
L
h h hg g g
WDT
0 1 2 2 1
1 2 1 1 0
...|| || || ... || ||
...
L L
L L L
g g g g g
h h h h h
0 1 10 1 2 12 2
0 1 1
, ,...,, ,...,
, ,...,n n
LD
L
h h hh h h
g g gWDT WDT
0 1 1 0 1 12 2, , , , , ,n nL Dh h h WDT WDT
Multiparametric Wavelet transform:
0 1 12, , ,n D WDT
Let be the smallest positive integer such that 2log 2m D 12 2 2m mD
1 1
1
0 1 1 0 1 12 2 212 , ,..., [ , ,..., ] IL Ln n r n n r
r n mh h h h h h
WDT AWT
Arbitrary cyclic wavelet transform, written in stairs-like form:
Structure of WDT
1 3
2 3 1
0 1 2 3 0 1
0 2
0
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0
2 3
2 3 0 1 0 1 2 3
0 1 2 3
2
1
13 0
g g g g
g g g g
g g g g
g g g g
g g
h h h h h h h h
h h h h h h h h
h h h h
h h h h
g g
g g g g
1
1
1
1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
=
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g
h h
g
h
g
h
=
16 0 1 2 3WDT h , h , h , h =
Stairs-Like Structure of WDT16(4)
stepstep
step
Two Spans:H-span and g-span
Two Spans:H-span and g-span
Two Spans:H-span and g-span
3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3 0 1 2 3
2 3 0 1 2 3 0 1
0 1 2 3 0 1 2 3
2 3 0 1 0 1 2 3
0 1 2
2 3 0 1
12 8
h h h h
h h h h
h h h h
h h h h h h h h
h h h h h h h h
g g g g g g g g
g g g g g g g g
g g g g
g g g g
I I =
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
=
Stairs-Like Structure of WDT16(4)
4 12 8 8 16 I IAWT AWT AWT
,
( ) ( )
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0( )
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
( ) ( )
0
i j
cos sin
si
1
1
1
1
1
1
1
1
n cos
i j
i
CS
j
.
Jacobi-Givens Rotations
1 1 0 0, , 1 , 0 0 1 12( ) ( ) ( ) , , ,n
k ki j k i j i j Lh h h CS CS CS AWT
Multiparametric presentation of AWT8(6)
0 1 2 3 4 5
0 1 2 3 4 5
4 5 0 1 2 3
2 3 4 5 0 18 0 1 2 3 4 5
5 4 3 2 1 0
5 4 3 2 1 0
1 0 5 4 3 2
3 2 1 0 5 4
, , , , ,
h h h h h hh h h h h h
h h h h h hh h h h h h
h h h h h h h h h h h hh h h h h h
h h h h h hh h h h h h
AWT
0,4 0 8 0 1 2 3 4 5
0 0 0 1 2 3 4 5
0 1 2 3 4 5
4 5 0 1 2 3
2 3 4 5 0 1
0 0 5 4 3 2 1 0
5 4 3 2 1 0
1 0 5 4 3 2
3 2 1 0 5 4
( ) , , , , ,
11
1
11
1
h h h h h h
c s h h h hh h h h h h
h h h h h hh h h h h h
s c h h h hh h h h h h
h h h h h hh h h h h h
CS AWTh h
h h
0 1 2 3
0 1 2 3 4 5
4 5 0 1 2 3
2 3 4 5 0 1
3 2 1 0
5 4 3 2 1 0
1 0 5 4 3 2
3 2 1 0 5 4
h h h hh h h h h h
h h h h h hh h h h h h
h h h hh h h h h h
h h h h h hh h h h h h
0 0
0 0
0 0cosc 0 0sins
0 5 0 0 0c h s h
0 0 0 5 0s h c h
3,7 0 2,6 0 1,5 0 0,4 0 8 0 1 2 3 4 5( ) ( ) ( ) ( ) , , , , ,h h h h h h CS CS CS CS AWT
0 0 0 1 2 3 4 5
0 0 0 1 2 3 4 5
0 0 4 5 0 1 2 3
0 0 2 3 4 5 0 1
0 0 5 4 3 2 1 0
0 0 5 4 3 2 1 0
0 0 1 0 5 4 3 2
0 0 3 2 1 0 5 4
c s h h h hc s h h h h
c s h h h hc s h h h h
s c h h h hs c h h h h
s c h h h hs c h h h h
h hh h
h hh h
h hh h
h hh h
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 18 0 1 2 3
3 2 1 0
3 2 1 0
1 0 3 2
3 2 1 0
, , ,
h h h hh h h h
h h h hh h h h
h h h hh h h hh h h h
h h h hh h h h
AWT
Multiparametric presentation of AWT8(6)
0,7 1 3,6 1 2,5 1 1,4 1 8 0 1 2 3 8 0 1( ) ( ) ( ) ( ) , , , ,h h h h h h CS CS CS CS AWT AWT
1,7 2 0,6 2 3,5 2 2,4 2 8 0 1 8( ) ( ) ( ) ( ) ,h h CS CS CS CS PAWT
1,7 2 0,6 2 3,5 2 2,4 2 0,7 1 3,6 1 2,5 1 1,4 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) CS CS CS CS CS CS CS CS
3,7 0 2,6 0 1,5 0 0,4 0 8 0 1 2 3 4 5 8( ) ( ) ( ) ( ) , , , , ,h h h h h h CS CS CS CS PAWT
Multiparametric presentation of AWT8
8 0 1 2 3 4 5 3,7 0 2,6 0 1,5 0 0,4 0, , , , , ( ) ( ) ( ) ( )h h h h h h CS CS CS CS AWT
0,7 1 3,6 1 2,5 1 1,4 1 1,7 2 0,6 2 3,5 2 2,4 2 8( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )CS CS CS CS CS CS CS CS P
08 0 3,7 0 2,6 0 1,5 0 0,4 0( ) ( ) ( ) ( ) T CS CS CS CS
18 1 0,7 1 3,6 1 2,5 1 1,4 1( ) ( ) ( ) ( ) T CS CS CS CS
28 2 1,7 2 0,6 2 3,5 2 2,4 2( ) ( ) ( ) ( ) T CS CS CS CS
0 1 28 0 8 1 8 2 8T T T P
Indexes of rotation matrices
0, 41, 52, 63, 7
4k k
4
1, 42, 53, 60, 7
1 4k k
4
2, 43, 50, 61, 7
2 4k k
08 0 3,7 0 2,6 0 1,5 0 0,4 0( ) ( ) ( ) ( ) T CS CS CS CS
18 1 0,7 1 3,6 1 2,5 1 1,4 1( ) ( ) ( ) ( ) T CS CS CS CS
28 2 1,7 2 0,6 2 3,5 2 2,4 2( ) ( ) ( ) ( ) T CS CS CS CS
General rule: 2
, 2n r
n rk i k
r – number of iteration within atomic function in multiparametric presentation i – number of matrices 2n
iiT
Multiparametric presentation of atomic
matrix
0
0 1 2 12 21
, , ,r ri
i Di D
h h hP T
AWT
1
0 1 2 1 2 20
, , , r r
Di
D ii
h h h T P
AWT
16 0 1 2 3 4 12 8 8 16, , ,h h h h I IWDT AWT AWT AWT
0 1 0 1 0 14 4 0 4 1 4 8 8 0 8 1 8 16 16 0 16 1 16, , T T P T T P T T PAWT AWT AWT
0 1 0 1 0 116 0 1 4 0 4 1 4 12 8 0 8 1 8 8 16 0 16 1 16, T T P I T T P I T T PWDT
1 1 1
4 4 12 8 8 8 16 160 0 0
( ) ( ) ( )i i ii i i
i i i
T P I T P I T P
2
1 1 1
1 0 1 2 1
1 1
0 1 12 2 2 2 2log 1 0
[ , ,..., ]2
[ , ,..., ] ( )n n r n r n n r
n r D
Di
D ir n L i
h h h
T P I
AWT
WDT
2
1 1
2
1 1 0
1, 2 2 2 2log 1 0 2 1
( )n r
n r n r n n rn r
D
D ik i kr n L i k
CS P I
Multiparametric presentation of WDT
Generalization:
Multiparametric presentation of wavelet
packets
2
1
0 1 2 1 1 0 1 2 1 12 2 2 2log 1
[ , ,..., ] [ , ,..., ]n D n r D n n rr n L
h h h h h h I
WDT AWT
16 0 1 2 3WDT h , h , h , h =
1 3
2 3 1
0 1 2 3 0 1
0 2
0
0 1 2 3
0 1
2 3
2 3 0 1 0 1 2 3
0 1 2 3
2 3 0
2 3
0 1 2 3
2 3
1
0 1
h h h h h h h h
h h h h h
g g g g
g g g g
h
g g g g
g g g g
g g g g
g g g
h h
h h h h
h h h h
g
1
1
1
1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
=
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2
0
3 0 1
1 2 3
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
h h h
g g g
h
g
Multiparametric presentation of wavelet
packets
16 0 1 2 3WDT h , h , h , h =
1 3
2 3 1
1 3
2 3 1
0 1 2 3 0 1 2 3
2 3 0 1 0 1 2 3
0 1 20 2
0
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
3
2 3 0
0 1 2 3
2 3 0 1
2
0
1
0
g g g g
g g g g
g g g g
g g g g
g g
h h h h h h h h
h h h
g g
g g g g
g g g g
g g g
h h h h h
h h h h
h h h h
h h
h h h h
g
h h
1
1
1
1
1
1
1
1
1 1
1 1
1 1
1 1
=
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1
0 1 2 3
2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g g
g g g g
g g g g
g
h h h h
g g g
g g g g
g g g g
g g g g
g g g g
2
1
0 1 2 1 1 0 1 2 1 12 2 2 2log 1
[ , ,..., ] [ , ,..., ]n D n r D n n rr n L
h h h h h h I
WDT AWT
Multiparametric presentation of wavelet
packets
16 0 1 2 3WDT h , h , h , h =
1 3
2 3 1
1 3
2 3 1
1 3
2 3 1
0 2
0
0 1 2 3
0 1 2
0 2
0
0
0 1 2
1
2
3 0 1 2 3
2 3 0 1 0 1 2 3
0 1 2 3
2 3 0
0 1 2 3
2 3 0 1
0 1 2 3
3
0
2 0 1
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g
h h h h h h h h
h h h h h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
h h h h
g g g
1
1
1
1=
3
0 1 2 3
2 3 0 1
0 1 2 3
0 1 2 3
0 1 2 3
2 3
0 1 2 3
0 1
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2
2 3
0 1 2 3
0 1 2 3
2
3
0 1 2 3
0 1 2 3
2 3
3 0 1
0 1
0 1
0
h h h h
h h h h
h h h h
h h h h
h h
g g g g
g g g g
g g g g
g g g g
g g g g
g g
h
h h
h h h h
h h h h
g
h h h
h h h h
h h h h
h h h h
h h h h
g
g
g
1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
g g g g
2
1
0 1 2 1 1 0 1 2 1 12 2 2 2log 1
[ , ,..., ] [ , ,..., ]n D n r D n n rr n L
h h h h h h I
WDT AWT
1 2 12( , ,..., ,..., )r r r r r
t rs s s s s 1
1
2
212
, 1,, 0.n r
n r
rtrt
rtn rs s
I s
AWTAWT
11 21 1 1 11
1
2 2 2 2 22
2...
rr rrr rtn n r n r n r n r
r
t
ss sss
AWP AWT AWT AWT AWT
1 1
0 1 2 11 1
1
1
1 2 11
11 21 1 1
1
1, ,...,
2 22
2 2 22
2[ , ,..., ]
...
Dr n m r n m
r n m
rn m r tn n r
rr r rn r n r n r
n
r
ts
ss s
h h hs s s s
WDP AWP AWT
AWT AWT AWT 1 11 ,ss
2 2 21 2, ,s ss
3 3 3 3 31 2 3 4, , , ,s s s ss 1 2 2, ,..., .n m n m n m n m
n ms s ss
Multiparametric presentation of wavelet
packets
Packet of atomic matrices:
1
01 1 12 2 2
( )D
ii
rt
n r n r n r
rtr
t sis
s T P
AWT
1 1
0 1 2 11 0
1 2 1
1 11
1, ,...,
2 2 2
2, ,..., ( )
D
D ir n m i
rn mt
n n r n r
rr tsi
t
sh h h φs s s T P
WDP
The inverse multiparametric wavelet
transform
2
1 1
1
2 2 2 2log 1
n n r n n rr n L
I
WDT AWT
2
1 1
2
log1
2 2 2 2 2 2 2log 1
n n r n n r r n r
t Lt t
r n L r n
I I
WDT AWT AWT
1
2 2 20
( )D
ir r i r
i
T P
AWT
( ) ( )t T T
0
2 2 21
( )t t ir r r i
i D
P T
AWT
0
2 , 22 1 2
( ) ( )ir i n r ik i k
n r n rk
T CS
12 1
12 , 20 12
( ) ( )r
ri
i r ik i kk r
T CS
21log 0 2 1
10 1 1 1 2 22 2 , 2
1 0 12
[ , ,..., ] ( ) n r
rLt
n D r r ik i kr n i D k r
P CS I
WDT
1
21 2 1
1
1
2
2log 2 10, ,..., 1
0 1 12 2 i, 21 1 0
[ , ,..., ] ( )r
rt
n mr
n r r
n r sL
tD ik ktr n i D k
s s s P CS
WDP
1r n r
Another form of multiparametric WDT
0 1 2 3 4 5
0 1 2 3 4 5
0 1 2 3 4 5
0 1 2 3 4 58 0 1 5
4 5 0 1 2 3
4 5 0 1 2 3
2 3 4 5 0 1
2 3 4 5 0 1
, , ,
h h h h h hg g g g g g
h h h h h hg g g g g g
h h h h h h h h hg g g g g gh h h h h hg g g g g g
CAT
8 0 1 5[ , ,..., ]||h h h
+ + + +
+ + + + + ++ + + + + +
+ + + + + ++ + + +
+ ++ + + + + +
+ + + + + ++ + + +
+
+ +
+
AWT 8 0 1 5[ , ,..., ]||h h h
é ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úê úë û
+ + + + + +
+ + + + + +
+ +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + ++ + + +
+ +
C
+ + + +
AT
1 1 1
0 1 2 3 4 2 2 2 10 2 1 2 1 2 2 1 2 1
r r
r r r r
The rule of permutation:
0 1 5 8 8 0 1 5, , , , , ,h h h h h hP 8AWT CAT
Sparse rotation matrix with reflection
, ( )
1 0 0 0
0 0
0 0
0 0 0 1
Ri j
i j
i
j
c s
s c
CS
0 1 2 12, , ,n Dh h h CAT
0 1 2 32, , ,n Dh h h CAT
01 0 23 0 2 2,2 1 0, , ,R R RD D CS CS CS
67 0 45 0 23 0 01 0 8 0 1 5
0 1 2 3
2 3 4 5
0 1 2 3
2 3 4 5
0 1 2 3
4 5 2 3
2 3 0 1
2 3 4 5
, , ,R R R R h h h
c s h h h hs c g g g g
c s h h h hs c g g g g
c s h h h hs c g g g g
c s h h h hs c g g g g
CS CS CS CS CAT4 5
0 1
4 5
0 1
4 5
0 1
4 5
0 1
h hg g
h hg g
h hg g
h hg g
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
2 3 0 1
2 3 0 1
0 1 2 3
8 0 1 2 3, , ,
g g g gh h h hg g g g
h h h hg g g g
h h h hg g g gh h h h
h h h h
CAT
Multiparametric presentation of CAT
70 1 56 1 34 1 12 1 8 0 1 2 3
2 3
0 1
2 3
0 1
2 3
2 3 0 1
2 3 0
0 1 2 3
, , ,R R R R h h h h
c s g gc s h hs c g g
c s h hs c g g
c s h hs c g g
s c h h
CS CS CS CS CAT0 1
2 3
0 1
2 3
0 1
2 3
0 1
2 3
g gh h
g gh h
g gh h
g gh h
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
8 0 1,
h hg g
h hg g
h hg g
h hg g
h h
CAT
01 2 67 2 45 2 23 2 8 0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
,
11
11
11
11
R R R R h h
c s h hs c g g
c s h hs c g g
c s h hs c g g
c s h hs c g g
CS CS CS CS
CAT
28C
Multiparametric presentation of CAT
12 12
12 12
12 12
12 12
12 12
12 12
12 12
12 12
c ss c
c ss c
c ss c
c ss c
-
-
-
-
+ + + +
+ + + +
+ + + +
+ +
+ ++ +
+ ++ +
+
+ + + +
++
+ ++ +
+ +
+ + +
+ + + ++ +
+ + + +
+ + + +
+ + + +
+ + + +
+ +
+ + + +
+ + + +
+ +
+
+ +
+ + + ++
12 12
12 12
12 12
12 12
12 12
12 12
12 12
12 12
c sc ss c
c ss c
c ss c
s c
-
-
--
+ +
+ +
+ +
+
+ +
+ +
+ +
+ ++
+ +
+ +
+ +
+ +
+ ++ +
+ ++ +
+
+ +
+ +
+ +
+ +
++ +
+ ++ +
12 12
12 12
12 12
12 12
12 12
12 12
12 12
12 12
c ss c
c ss c
c ss c
c ss c
-
-
-
-
+
+
+
+
+
+
+
++
++
++
+ ++
11
11
11
11
--
--
--
--
2 1 0 28 2 8 1 8 0 8 0 1 5 8, , ,h h h T T T CCAT
28 2 01 2 67 2 45 2 23 2
18 1 70 1 56 1 34 1 12 1
08 0 67 0 45 0 23 0 01 0
,
,
.
R R R R
R R R R
R R R R
T CS CS CS CS
T CS CS CS CS
T CS CS CS CS
2 21
n ni i
i i T T
8 0 1 5 8 0 1 2
0 1 2 28 0 8 1 8 2 8
, , , , ,1h h h
T T T C
CAT CAT
8 0 1 5 8 0 1 2
0 1 2 28 8 0 8 1 8 2 8
, , , , ,
1
h h h
P T T T C
AWT AWT
16 0 1 5 8 0 1 5 8 16 0 1 5
0 1 2 2 0 1 2 28 8 0 8 1 8 2 8 8 16 16 0 16 1 16 2 16
, , , , , , , , ,
1 1
h h h h h h h h hI
P T T T C I P T T T C
WDT AWT AWT
Flowchart of WDT16
1 2 2
0 1 2 1 0 1 12 2
1 1 01 1
2 , 2 12 2 2 2 20 0 2 1
, , , , , ,
1 1 ,
n n
n n n n nn nn
D D
D Di D R D
i ii k i ki i k
h h h
P T C P CS C
AWT AWT
1 1 1
1 11
0 1 1 12 2 22 2 21 0
[ , ,..., ] 1 n r n r n r
Di D
n D i n n rr n m i
P T C I
WDT
1 11 12 2
1 1 01 1
12 , 2 12 2 2 21 0 2 1
1 ,n r n rn r n rn r
Dn m R D
i n n ri k i kr n m i k
P CS C I
1 2 1
1 1 1
1 11 12 2
21 1, ,..., 1
0 1 1 2 2 22 11 0
2 1 01
2 , 2 12 21 0 2 1
[ , ,..., ] 1
1
tr
n m
n r n r n r
n r n rn r n rn r
n
r sDi D
D itr n m i
rD
n m R Dii k i kt i k
s s s P T C
P CS C
WDP
11
1
.
trs
r n m
Multiparametric presentation of WT
Multiparametric presentation of atomic
matrix
The inverse multiparametric wavelet
transform
1 1
10 1 1 0 1 12 2
1 1 11 1
12 2 2 2 2 2 2 2 20 0 0
, , , , , ,
1 1 1
n n
n n n n n n n n nD D i
tD D
t tD D Dt t ti D i t D ti i D i
i i i
P T C C T P C T P
AWT AWT
111 12 2n n
D itD iD i D iT T
2
2 2
1log 1
11 10 1 1 12 2 2 2 22
0
1 2 11 1
11 2 , 1 2 12 2 2 20 0
[ , ,..., ] 1
1
r r r n r
n r
r r n rr r
D iL Dtn m D t
n D D ir n i
Dtn m D R tD iD i k D i k
r n i k
C T P I
C CS P I
WDT
2log L
1 2 1
2
2 2
, ,..., 10 1 12
2log 1 2 11 1
11 2 , 1 2 12 21 0 0
[ , ,..., ]
1
n m
trn r
r rr r
n D
n r sL Dtn m D R tD iD i k D i ktr n i k
s s s
C CS P
WDP
11
10 1 1 12 2 2 2
0
, , , 1 .n n n nD i
Dtt D tD D i
i
C T P
AWT
Compression properties estimation
1 2,E
2 1
QUESTIONS?
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