!"#"$%&' (&)**$&+ ,$'!-#* ,"# %-!-.!$"+ &+% -/!#&.!$"+ ", 0#"0-#!$-* ", *0$.)'&!-% 1&**-* Mehul P. Sampat1 , Alan C. Bovik1 , Mia K. Markey1 , Gary :. Whitman2 , Tanya W. Stephens2 !"# $%&'#()&*+ ,- !#./) /* 01)*&%2 01)*&%2 !3 454672 $80 !"# $%&'#()&*+ ,- !#./) 9: ;: 0%<#(),% =/%>#( =#%*#(2 ?,1)*,%2 !3 44@A@2 $80 1

2

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

'#()((#*(+%#,-*+-.)*/**"0)**+"!111

>/%>#() OA\NP: =,GF1*#(E0&<#< ;#*#>*&,% S=0;#T )+)*#G) "/'# J##% <#'#D,F#< *, /&< (/<&,D,H&)*) &% <#*#>*&%H G/GE G,H(/F"&> D#)&,%) *"/* G/+ &%<&>/*# *"# F(#)#%># ,- J(#/)* >/%>#( O]\5P: !"#)# )+)*#G) />* /) / )#>,%< (#/<#( /%< *"# !%/D <#>&)&,% &) G/<# J+ *"# (/<&,D,H&)*: 9,)* )*1<&#) "/'# )",C% *"/* =0;# )+)*#G)2 C"#% 1)#< /) /% /&<2 "/'# &GE F(,'#< (/<&,D,H&)*)^ />>1(/>+ &% *"# <#*#>*&,% ,- J(#/)* >/%E >#( O_P: 9/))#) C&*" )F&>1D/*#< G/(H&%) >/((+ / G1>" "&H"#( (&)K ,- G/D&H%/%>+ *"/% ,*"#( *+F#) ,- G/))#) ,( >/D>&!>/E *&,%): ?,C#'#(2 >1((#%* =0;# )+)*#G) /(# <(/G/*&>/DD+ J#*E *#( /* <#*#>*&%H G&>(, >/D>&!>/*&,%) *"/% G/))#): B# "/'# J##% <#'#D,F&%H %#C G#*",<) -,( *"# <#*#>*&,% ,- )F&>1D/*#< G/))#) S89T /%< /(>"&*#>*1(/D <&)*,(*&,%) S0;T: I,( *"&) */)K2 C# "/'# <#'#D,F#< / *C, )*#F /DH,(&*"G -,( *"# <#*#>*&,% ,- )F&>1D/*#< D#)&,%): L% *"# !()* )*/H#2 C# #%"/%># )F&>1D/*&,%) ,% G/GG,H(/G) 1)&%H / %#C #%"/%>#G#%* /DE H,(&*"G: !"&) #%*/&D) !D*#(&%H &% *"# X/<,% <,G/&%: !"# H,/D ,- *"# )#>,%< )*/H# &) *, <#*#>* *"# )F/*&/D D,>/*&,%) C"#(# *"# )F&>1D#) >,%'#(H#: I,( *"&) F1(F,)#2 C# "/'# &%'#%*#< / %#C >D/)) ,- !D*#() /%< !D*#( J/%K) *"/* /(# )F#>&!>/DD+ *1%#< *, G/*>" *"# F"+)&>/D )*(1>*1(#) *"/* <#!%# )F&>1D/*#< D#)&,%): B# >/DD *"#)# /) 8F&>1D/*&,% !D*#() S8I)T /%< 8F&>1D/*&,% !D*#( J/%K) S8I`)T: !"#)# !D*#() /(# %/((,C J/%< !D*#()2 C"&>" /(# <#)&H%#< *, G/*>" *"# F"+)&>/D )*(1>*1(#) ,- *"#)# D#)&,%) /%< -,(G / %#C >D/)) ,- C/'#D#*E*+F# !D*#( J/%K): 0 K#+ /)F#>* ,*"&) C,(K &) *"/* *"# F/(/G#*#() ,- *"# <#*#>*&,% /DH,(&*"G /(# J/)#< ,% G#/)1(#G#%*) ,- F"+)&>/D F(,F#(*&#) ,- )F&>1D/*#< D#)&,%): !"#)# G#/)1(#G#%*) C#(# G/<# J+ *C, #.F#(&#%>#< (/<&,D,H&)*) SabB2 !B8T: 0 <#*/&D#< <#)>(&F*&,% ,- *"&) /DH,E (&*"G >/% J# -,1%< &% O6@2 66P: !"# H,/D) ,- *"&) F/F#( /(# /) -,DD,C): c(#'&,1)D+2 C# "/'# -,>1)#< ,% *"# <#*#>*&,% ,- *"# (#H&,% C"#(# *"# )F&>1D#) >,%'#(H#: L% *"&) F/F#( C# )",C *"/* *"# H/1))&/% *,(,&/% J# 1)#< *, <#*#>* *"# >#%E *(/D G/)) (#H&,% ,- 89: 8#>,%/D F(,F#(*&#) ,- 89 >/% J# #.*(/>*#< -(,G *"# (#)F,%)#) ,- *"#)# !D*#(): !, '#(&-+ *"# />>1(/>+ ,- *"#)# /1*,G/*&>/DD+ #.*(/>*#< '/D1#)2 C# >,GF/(#< *"#G *, *"# G#/)1(#G#%*) G/<# J+ *"# (/<&,D,H&)*)2 1)&%H / )*/*&)*&>/D *#)* ,- #U1&'/D#%>#:

!!"#"$%&

!23445")**+

5. !6-"#7

:. 1-!6"%*

L% *"&) )#>*&,% C# F(#)#%* / (#'&#C ,- *"# 8I): 0 8I >/% J# <#)>(&J#< /) / a/1))&/%EG,<1D/*#< )&%# *,(1): !"# 8I >,%)&)*) ,- U1/<(/*1(# >,GF,%#%*) C"&>" /(# *"# >,)&%# 8I Sfcosine (r, θ; r0 , σ, ω)T /%< *"# )&%# 8I Sfsine (r, θ; r0 , σ, ω)T: !"#)# >,GF,%#%*) /(# )",C% &% I&H: 6 /%< /(# <#!%#< /) -,DD,C)d

L% *"&) )#>*&,%2 C# <#)>(&J# *"# G#*",<,D,H+ 1)#< *, <#*#>* *"# >#%*(/D G/)) (#H&,%) ,- 89 ,% *"# >,GFD#*# G/GG,H(/G: B# /D), <#)>(&J# *"# G#*",<) *, #.*(/>* *"# F(,F#(*&#) ,- 89 -(,G (#H&,%) ,- &%*#(#)* SXWL)T >,%*/&%&%H 89:

S6T

!"# &G/H#) -,( *"&) )*1<+ C#(# ,J*/&%#< -(,G *"# ;&H&*/D ;/*/E J/)# -,( 8>(##%&%H 9/GG,H(/F"+ S;;89T O67P: !"# ;;89 &) *"# D/(H#)* F1JD&>D+ /'/&D/JD# ,%)&)*) ,- 72]7@ >/)#): Q/>" >/)# >,%)&)*) ,- -,1( G/GG,H(/G)d / == /%< 9VW '&#C ,- #/>" J(#/)*: !"# ,1*E D&%#) ,- *"# /J%,(G/D&*&#) /(# )*,(#< &% e>"/&% >,<#f /%< /(# /'/&D/JD# &% *"# "/&% >,<#f2 J,(<#() ,- *"# /J%,(G/D&*&#) >/% J# (#>,%)*(1>*#<: 0-*#( (#>,%)*(1>*E &%H *"# J,(<#(2 *"# &%*#(&,( ,- *"# ,1*D&%# &) !DD#< *, >(#/*# / eG/)Kf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cL C#(# FD,**#< *, ,J*/&% *"# IXW= >1('#: !, #'/D1/*# *"# F#(-,(G/%># ,- *"# >#%*(/D G/)) (#H&,% <#E *#>*&,% /DH,(&*"G2 / )#* ,- &G/H#) >,%)&)*&%H ,- MN &G/H#) ,89 C/) 1)#<: !"#)# &G/H#) C#(# )>/%%#< C&*" / )&%HD# <&HE &*&Z#(2 >,%*/&%#< / )&%HD# D#)&,% /%< C#(# (/%<,GD+ )#D#>*#<: !"# (/<&,D,H&)*) SabB /%< !B8T "/< G#/)1(#< *"# F(,F#(E *&#) ,% / <&--#(#%* )#* ,- 67 89: !"&) )#* C/) 1)#< *, #.*(/>* *"# F(,F#(*&#) ,- *"# 89 -(,G *"# !D*#( (#)F,%)#):

fcosine (r, θ; r0 , σ, ω) = g(r; r0 , σ) ∗ cos(ωθ)

fsine (r, θ; r0 , σ, ω) = g(r; r0 , σ) ∗ sin(ωθ) S7T 2 2 SAT g(r; r0 , σ) = exp[−(r − r0 ) /(2 ∗ σ )] ! C"#(#d r = x2 + y 2 , θ = arctan(y/x) /%< σ &) *"# )*/%E + G#/)1(#< &% >+>D#) F#( >&(>1G-#(#%>#: !"# a/1))&/% *,(1) &) <#!%#< J+ #U1/*&,% SAT: !"# >,GFD#. 8I -,(G / %#C >D/)) ,- U1/<(/*1(# !D*#() /%< "/'# *"# /<'/%*/H# *"/* *"# )1G ,*"# )U1/(#< (#)F,%)#) ,- *"# *C, U1/<(/*1(# >,GF,%#%*) &) F"/)# &%<#F#%<#%*: 0 <#*/&D#< <#)>(&F*&,% ,- *"# 8I) /%< !DE *#( J/%K) >/% J# ,J*/&%#< &% O6@2 66P:

S/T =,)&%# 8I

SJT 8&%# 8I

,89. 3: Q./GFD# ,- / 8F&>1D/*&,% !D*#( S8ITd I&H1(#) S/T /%< SJT )",C *"# *C, U1/<(/*1(# >,GF,%#%*) ,- *"# 8I2 %/G#D+2 *"# >,)&%# 8I /%< *"# )&%# 8I2 (#)F#>*&'#D+:

:.3. %;<; %=>?@8A<8BC

:.5. 1=;>D@=E=C<> BF A@BA=@<8=> GH @;I8BJB98><> !"# F"+)&>/D F(,F#(*&#) ,- 89 G#/)1(#< J+ *"# (/<&,D,H&)*) SabB /%< !B8T C#(#d *"# G/Y,( /.&) ,- *"# >#%*(/D G/)) (#H&,%2 *"# %1GJ#( ,- )F&>1D#) /%< *"# D#%H*" /%< C&<*" ,)F&>1D#): !"#)# G#/)1(#G#%*) C#(# G/<# ,% (#H&,%)E,-E&%*#(#)* SXWL)T >,%*/&%&%H / 89: B# 1)#< *"# XWL 9/%/H#( FD1H&% ,- gL? LG/H#b S"**Fdhh()J:&%-,:%&":H,'h&YhT *, /DD,C / 1)#( *, G/K# *"#)# G#/)1(#G#%*): 0 )*/*&)*&>/D *#)* -,( #U1&'/D#%># C/) /FFD&#< /%< &* C/) )",C% *"/* *"# G#/)1(#G#%*) ,- *"# *C, (/<&,D,H&)*) C#(# #U1&'/D#%* O6AP: S/T !,(,&
SJT 6)* <#(&'/*&'# ,*,(,&
:.:. %=<=?<8BC BF ?=C<@;J E;>> @=98BC BF *1

,89. 5: I&H: S/T )",C) *"# *,(,&*#< &% I&H: SJT:

c(#'&,1)D+2 C# "/'# 1)#< *"# 8I) -,( *"# <#*#>*&,% ,- 892 C"&>" /(# G/(K#< J+ >,%'#(H&%H D&%#) ,( )F&>1D/*&,%): I,( *"# <#*#>*&,% ,- *"# >#%*(/D G/)) (#H&,% ,- 89 /%< *, #.*(/>*

!!"#"$%(

Filter output at center of image

*"# &%-,(G/*&,% /J,1* *"# >#%*(/D G/)) (#H&,% ,- / 89 C# 1)# *"# !()* <#(&'/*&'# ,- *"# H/1))&/% *,(1): V#* g ! (r; r0 , σ) <#%,*# *"# !()* <#(&'/*&'# ,- *"# *,(,&
2

2

2

Q/>" &G/H# &) !D*#(#< C&*" *"# !D*#( Sg ! (r; r0 , σ)T: L- *"# !D*#( eG/*>"#)f / 892 / F#/K &) ,J*/&%#< /* *"# )F/*&/D D,>/*&,% >,(E (#)F,%<&%H *, *"# >#%*#( ,- *"# 89: !"1)2 )1)F&>&,1) (#H&,%) >/% J# &<#%*&!#< J+ <#*#>*&%H *"# D,>/D F#/K) &% *"# ,'#(/DD ,1*F1*: !"# D/(H#( *"# ,1*F1* /* / F/(*&>1D/( )F/*&/D D,>/*&,%2 *"# "&H"#( *"# D&K#D&",,< *"/* *"# )F/*&/D D,>/*&,% >,((#)F,%<) *, *"# >#%*#( ,- / 89: 8&%># G/))#) >/% J# ,- '/(&,1) )&Z#)2 C# !()* !D*#( *"# &G/H# C&*" / )#* ,- !D*#() C&*" <&--#(#%* (/<&& r0 : I,( #/>" F&.#D2 C# !%< *"# G/.&G1G (#)F,%)# />(,)) *"# )#* ,- !D*#(#< &G/H#): !"1)2 C# ,J*/&% / )&%HD# &G/H# C"#(# *"# &%*#%)&*+ /* #/>" F&.#D <#%,*#) *"# G/.&G1G (#)F,%)# ,J*/&%#< /* *"/* F/(*&>1D/( F&.#D: I&%/DD+2 C# !%< *"# *,F N G/.&G/ &% *"&) &G/H#: 0 F&.#D &) ,%D+ G/(K#< /) / F,*#%*&/D G/))2 &- &* C/) G,(# *"/% / !.#< <&)*/%># -(,G F(#'&,1)D+ G/(K#< F,&%*)R *"# <&)*/%># C/) )#* J+ *"# /'#(/H# )&Z# ,- / 89 G#/)1(#< &% ,1( G#/)1(#G#%* )*1<+:

180 160 140 120 100 80 60 40 20 0 !20 90

Radius of idealized mass = 140 pixels

100

110

120

130

140

150

160

170

180

Radius of idealized mass = 140 pixels

SJT I&D*#( W1*F1*)

S/T L<#/D&Z#< G/))

,89. :: I&H: S/T )",C) *"# /% &<#/D&Z#< G/)) S(/<&1) i6M@ F&.E #D)T C&*" /<<&*&'# H/1))&/% %,&)#: !"# (#)F,%)# ,- *"# !D*#() S/* *"# >#%*#( ,- *"# &G/H#T ,- <&--#(#%* (/<&& &) <#F&>*#< &% I&H: SJT: !"# G/.&G1G (#)F,%)# &) ,J*/&%#< C"#% *"# !D*#( (/<&1) &) G/*>"#< *, *"# )&Z# ,- *"# &<#/D&Z#< G/)):

&G/H&%H: !"1) ,%# >/% (#Y#>* *"# %1DD "+F,*"#)&) SH0 T2 &*"# FE'/D1# ,J*/&%#< -,( *"&) )*/*&)*&>/D *#)* &) D#)) *"/% @:@N: 0D),2 %,*# *"/* )G/DD#( '/D1#) ,- δ &GFD+ / )*(&>*#( >(&*#(&/ -,( />"&#'&%H #U1&'/D#%>#: K. #-*)'!*

:.K. -L<@;?<8BC BF AMH>8?;J A@BA=@<8=> BF *1

C"#(# x i *"# G#/% ,- *"# <&--#(#%>#) J#*C##% *"# G#/)1(#E G#%*) ,- *"# (/<&,D,H&)* /%< *"# /DH,(&*"G: s i )*/%#) J#*C##% *"# G#/E )1(#G#%* ,- *"# (/<&,D,H&)* /%< *"# /DH,(&*"G: δ i @:A S,( @:7NT × G#/% ,- *"# (/<&,D,H&)*^) G#/)1(#G#%*: !"# -/>*,( δ />>,1%*) -,( *"# #.F#>*#< '/(&/J&D&*+2 J/)#< ,% ,1( K%,CD#/D

!"# (#)1D*) ,- *"# <#*#>*&,% ,- *"# >#%*(/D G/)) (#H&,% /(# F(#E )#%*#< &% *"# -,(G ,- / IXW= >1('#: !"&) &) )",C% &% I&H: M: 0 )#%)&*&'&*+ ,- _@[ C/) />"&#'#< /* 7@ IcL: I,( *"# #.*(/>E *&,% ,- *"# F/(/G#*#() ,- *"# >#%*(/D G/)) (#H&,%2 / )*/*&)*&>/D *#)* ,- #U1&'/D#%># C/) 1)#< *, >,GF/(# *"# (#)1D*) ,- *"# /DE H,(&*"G /%< *"# G#/)1(#G#%*) G/<# J+ *"# (/<&,D,H&)*): !/E JD# 6 )",C) *"#)# (#)1D*): B# )",C *"/* *"# G#/)1(#G#%*) /(# #U1&'/D#%* -,( (#/),%/JD# >",&>#) ,- *"# F/(/G#*#( δ:

1 0.9 0.8 0.7

Sensitivity

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# False Positives per Image

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toroidal gaussian filters for detection and extraction of ...

Screening mammography, which is x4ray imaging of the breast, is ... of quadrature filters and have the advantage that the sum of the squared .... imaging. Thus one can reject the null hypothesis (H0), if the p4value obtained for this statistical test is less than 0.05. Also, note that smaller values of δ imply a stricter criteria for.

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