InverseProblemsin SpacePhysics
BerndInhester, May 2002
The program
Part I: ExamplesImagedeblurringTomographyRadiative transferinversionHelioseismology
Part II: Mainly directmethodsFouriertransformsSingularvaluedecompositionBackus-Gilbertor Mollifier method
Part III: Mainly IterativemethodsNoiseandapriori knowledgeIterationalgorithmsRegularizationby TikhonovNonlinearproblemsGencodeandNeuralnetworks
InverseProblemsin SpacePhysicsII/0 May 2002
FT-inversion: Convolution kernels
Convolution problemsof the type (including generalizationstohigherdimensions,
�is thesizeof thedomain)
��������
� ���� ���������������������
canbesolvedin principleby Fouriertransform.Wedefine��������� ��
� �"!$#&% ����������� ' ��������#
�(!$#&% �����)���"� ' � �*,+ - .
An FT of the convolution type integral equationyields a simplealgebraicrelationbetweentherespectiveFouriercoefficients������)��� � �� ����� ������)�We thereforehave a problemfor which an anlytic inversionfor-mulaexists ��������
#� !�#&% ��������
�� ���)�Otherinversionproblemscanbe broughtinto the form of a con-volutionproblemby meansof variabletransforms.Thesolarlimbequationis anexamplefor a kernelof thedivision type
��������/
0� � �� � ���������1���"���
Use�2� �(3 , � � � �(3&4 , �"� � � �5364 ��7 � to obtain
�������8�9;:�/
9;: 0� � � 3 � 364 �< =?> @ ��� � 3&4 � � 364< =?> @ �"7��� � ��7A 7 � � � � ��7 � �
Many otherinverseproblemshaveanalyticinversionformulas.
InverseProblemsin SpacePhysicsII/1 May 2002
FT-inversion: X-ray transform
WehavenotedthattheX-ray transformis alsocloseto a convolu-tion typeproblem
�Bwecanalsoapproachit by FT
X-ray transform:����CD'FE�GH��� I
� I���JC K LME�GN�O�PL
Inserttheinverse3D Fouriertransformof�
����CQ��R � ! RMSUT �����VO�W' V � +
XZY ' Y - .�[into theX-ray transformfor fixed
E GI� I
���JC K LME�GN���PL\�R
�����VO� I� I
� ! R,S^]_Ta`�b�cedgf ��L< =?> @
h * XRjikced � !
R,SlT ����mVO�* X � ! RMSUTon&p$qsr � X VutJE G �
This is exactlyof theform of a2D FT in theimageplane.
Hence����Vv'5E�GH�
=* X ����mVO�
forV
in theplane w E�G.
xzy|{~}��|�5�Jy|������y|�^�$���zy|��{
x�
y�z�
k�
x�
k�
y�kz�
���
������� ������
� �¡�����{£¢��U¤¥{��¦ �~�§ § §¨ ¨
�Fy©�zª�«¬~�®�zy|¢��|�F��y|���¯�~y©�^�$���zy|��{�x
kx
ky�kz
Illustrationof theFourier reconstructionof theX-ray transform
InverseProblemsin SpacePhysicsII/2 May 2002
FT-inversion: The noiseproblem
Thepracticalusefulnesof analyticinversionformulasis limited ifthedatais contaminatedwith noise
��������
� ���� �������������1���"���°K ±?�����
then �����)��� � �� ���)� ����m����K �± ���)�Thenoiseis assumedof zeromeanmeanandcorrelationlength
�³².
Thenits Fouriercoefficients�±?���)�
arerandomcomplex numbersofzeromeanandvariance
� �³²m´µ�v¶ �· ± ¶ ���independentof
�aslong
ask ¸ 2+ ´µ� ²
.
For theapplicationof theanalyticinversionformulathenoiseis adesaster:
��������K � ² �������#
� !$#&% ���m����� ���)� K #
� !$#6% �± ���)��� �m���
w¹ aº v» e¼ n½ u¾ m¿ bÀ
e¼ rÁ
lo
 gà pÄ o
 wÅ eÆ r sÇ pÄ eÆ c
È tÉ ru
Ê m
kË eÌ rÍ nÎ eÌ lÏ 2
dÐ aÑ tÒ aÑ 2+Ó 2
lÏ oÔ gÕ (Ö 1× )Ø 2
noÔ isÙ eÌlÏ eÌ vÚ eÌ lÏ lÏ oÔ gÕ 2
2
maÛ x
w¹ aº v» e¼ n½ u¾ m¿ bÀe¼ rÁ
moÜ dÝ e¼ l 2Þ 2
mß aà xá
Power spectra of data â ãåäçæ�è (solid) andkernel é (dashed)beforeandafter theinversion,i.e., divisionby êé
InverseProblemsin SpacePhysicsII/3 May 2002
FT-inversion: Truncated spectrum
We have to limit the spectrumto wave numbers�aëíìíî :ðï ¸ �òñ�ómô
where�añ�ómô
is givenby theintersectionof thenoiselevel with thekernelspectralpower whenboth,dataandkernelspectraarenor-malizedto thesamevalueat
�= 0�� �Jõa�
�� ��� ñ�ómô � � �� ¶ ��õò�öK �± ¶ ��õa��± ¶ ��õa� � � K ÷�øúù
Thesignal-to-noiseratio is heredefinedas
÷søúù � �� ¶ ��õa��± ¶ ��õa� � � � ´µ�o¶ �5��· �û����� ¶
� � ² ´µ� ¶ � · ± ¶ �"�As anexample:imagedeblurringwith Gaussiankernelin 1D
� ����� h �Hü�ý 2�* ���³þ � ¶ ' �� ���)� h �Hü�ý ÿ�* �m� � þ � ¶
then � q �� ��õa��� ���añ�ómôF� h �* ��� ñ�ómô � þ � ¶ � *+ ¶�� � þ� � ¶�� ¶ñ�ómô� ñ�ómô
is the maximumnumberof complex Fourier coefficientsofthereconstruction. �B
the numberof independentimageparametersof the recon-struction(factor2 becausetheFouriercoefficientsarecomplex)
* � ñ�ó�ô h �� þ�� � q � � K ÷�øúù �
where�Q´µ� þ
are the numberof independentimage parameterswhich have beenmeasured.With theknowledgeof thekernelwecanenhancethenumberof independentimageparametersdepend-ing onSNR.
InverseProblemsin SpacePhysicsII/4 May 2002
FT-inversion: Example with little noise
-1� . 0 -0 . 5� 0 . 0 0 . 5� 1� . 0x
0 . 00 . 20 . 4�0 . 6 0 . 8�1� . 0s� i� g� n� a� l� s
� t� r� e� n� g� t� h�o� b�
s� e� rv� e� d�o� r� i g! i
n" a# l$
0 5� 1� 0 1� 5� 2% 0 2% 5� 3& 0w' a# v� e� nu( mb
�e� r-9)-6
-3&03&
l� o* g� p+ o* w, e� r
�o� b� s� e� r� v� e� d�
k-
e� r� n" e� l$ s� p. e� c/ t0 r� u( m1n" o� i s� e�l$e� v� e� l$
k-
ma2 xk-
t3ru4 nc5
r� e� c/ o� n" s� t0 r� u( c/ t0 e� d�
-1� . 0 -0 . 5� 0 . 0 0 . 5� 1� . 0x
0 . 00 . 3&0 . 6 0 . 9)
s� i� g� n� a� l� s
� t� r� e� n� g� t� h�o� b�
s� e� r� v� e� d�re� c/ o� ns� t0 ru( c/ t0 e� d�o� r� i g! i
n" a# l$
Exampleof a reconstruction.TheGaussiankernelhasjusta widthcorrespond-ing to the distanceof the two peaksin the original signal. Noisevarianceis0.0056 , thespectrumis truncatedat 3/4 798;:=< .InverseProblemsin SpacePhysicsII/5 May 2002
FT-inversion: Example with more noise
-1� . 0 -0 . 5� 0 . 0 0 . 5� 1� . 0x
0 . 00 . 20 . 4�0 . 6 0 . 8�1� . 0s� i� g� n� a� l� s
� t� r� e� n� g� t� h�o� b�
s� e� r� v� e� d�o� r� i g! i
n" a# l$
0 5� 1� 0 1� 5� 2% 0 2% 5� 3& 0w' a# v� e� nu( mb
�e� r-9)-6
-3&03&
l� o* g� p+ o* w, e� r
�o� b� s� e� r� v� e� d�
k-
e� r� n" e� l$ s� p. e� c/ t0 r� u( m1no� is� e�l$e� v� e� l$
k-
ma2 xk-
t3ru4 nc5
re� c/ o� ns� t0 ru( c/ t0 e� d�
-1� . 0 -0 . 5� 0 . 0 0 . 5� 1� . 0x
0 . 00 . 3&0 . 6 0 . 9)
s� i� g� n� a� l� s
� t� r� e� n� g� t� h�o� b�
s� e� rv� e� d� re� c/ o� ns� t0 ru( c/ t0 e� d�o� r� i g! i n" a# l$
Exampleof a reconstruction.TheGaussiankernelhasjusta widthcorrespond-ing to the distanceof the two peaksin the original signal. Noisevarianceis0.056 , thespectrumis truncatedat 2/3 7>8?:=< .InverseProblemsin SpacePhysicsII/6 May 2002
SVD-inversion: The basics
Weneedageneralizationof FT for moregeneralinverseproblems
�������� � ���O'H�������������1���"���A@CBD � E F ' F -HGJI 'KD - GJLThebasicideais to constructa symmetricmatrix from
Ewhich
hasacompleteorthogonalsetof eigenvectorsandrealeigenvaluesõ EE M õ �ON !P ! � � Q ! �RN !P ! � or
E P ! � Q ! N !E M N ! � Q ! P !Thereare S vectorsof the P ! which orthogonallyspanthe mod-elspaceG I and T vectorsof the N ! which orthogonallyspanthedataspaceG L becauseE M E P ! � Q ¶! P ! ' E E M N ! � Q ¶! N ! �B
For everynonzeroQ ! thereis alsoanegativeone.Thereareat
most U :WVYX ì[Z =min(T ,S ) pairsof nonzeroeigenvalues\ Q ! �BTheactionof
Eis completelydescribedby its singularvalue
decomposition(to bereadasa dyad)E � !^]`_badcfe!hgji N ! Q ! P ! ' P ! - G I ' N ! -HG L
whereP ! and N ! arenormalizedto unity andallQ ! chosenpositive
andorderedsothatQ ilk Q ¶ k mnmom Q !^]p_dadcfeKk õ
.q ThedecompositionE B r P ! ' N ! 'sQ !Yt U � � ' U :WVuX ìbZ canbefound
numericallyfor U :WVYX ì[Z ¸ about1000.
InverseProblemsin SpacePhysicsII/7 May 2002
SVD-inversion: The nullspace
In mostcasesU :WVYX ì[Z ¸ both S and T .
If U :WVYX ì[Z ¸ S not all model featuresaremappedinto dataspace.Therearenonequalmodels
Fwhich cannotbe distinguishedby
theobservationoperationofE
. Thespacespannedby P ! withQ !
= 0 is thenullspacev �wE �of theobservationoperator.
If U :WVYX ì[Z ¸ T the observationsdo not cover the total dataspace.Thereareinconsistentvectors
Dwhich canimpossiblybe the re-
sult of anobservationthroughE
. Thespacespannedby N ! withQ !yx� 0 is therangez �wE �of observationoperator.
As ageneralizedinversetoE
we defineE �{i|~}�� � !^]p_dadcfe!hgji P ! �Q ! N !
1
2
(�
)�
ra� ng� e�(�
)�
1
2
N�
u� l�l�s� p� a� c� e� (
�)�
(�
)�
i�n� c� o� n� s� i� s� t� e� n� t
�
(� 1� )�ind� is� t� ing� u� is� ha� b� le� (� 2)�(� 3� )�
- 1�D�
A�
T�
A�
S� P�
A�
C� E�
MO� DEL S� PAC� E
Mappingof � anditsgeneralizedinverse� �¡ ¢¤£n¥ betweendataandmodelspace.The“visible” part of modelspaceis ¦ â[� èw§ , theorthogonalcomplementof thenull space¦ â[� èInverseProblemsin SpacePhysicsII/8 May 2002
SVD-inversion: The noiseproblem
If noiseis added,the dataD K ¨
is almostcertainlyinconsistent.Weassumethenoise
¨to havezeromeanandvariance© ¶² .E �{i|~}�� ignoresthe part of the noisewhich falls out of the rangez �¤E �
. Yet theeigenvaluescloseto zeroareproblematic:F K F ² � E �{i|~}�� �pD K ¨ò�� !ª]`_dabcªe!hgji P !¬« � N ! t¤DW�Q ! K � N ! tu¨ò�Q !
because� N ! tu¨ò� arerandomrealnumberswith zeromeanandvari-
ance© ¶² (the N ! arenormalized).
�BWehavethesameproblemasin Fourierinversion.Depending
onthenoiselevel,wehaveto truncatethespectrumofE �{i|®}A� below
modeU ñ�ómô to bedeterminedfromQ iQ ! 8;:¯< h � � N i twDW� ¶ K © ¶²© ²
m° o± d²e³ n´ uµ m° b
¶e³ r· i¸
a¹ bº s» mo¼ d½ e¾ a¹ mp¿ litÀ uÁ d½ e¾
i nà zÄ eÅ rÆ oÇkeÈ rneÈ lsÉ pÊ eÈ cË tÌ ruÍ m
(Î )ÏnoÐ isÉ eÈleÈ vÑ eÈ l (Î )Ïi maÒ x
m° o± d²e³ n´ uµ m° b
¶e³ r· i¸
(Î )Ï (Î )ÏimaÒ x
NormalizedSVDspectra of data â�ã äQæ è (solid)andkernel é (dashed)beforeandafter theinversion,i.e., divisionby ÓÕÔ
InverseProblemsin SpacePhysicsII/9 May 2002
SVD-inversion: 2D tomography, model and data
OÖ r× iØ gÙ iØnÚ aÛ lÜ mÝ oÞ dß eà lÜ
Tomographygrid andmodeldensity. Thegrid is cylindrical with á asazimuthangleand â asdistance.
dã aä tå aä wæ iç tå hè 1é pê eë rì cí eë nî tå nî oï iç sð eë
-1.ñ 0ò -0ò.ñ 5ó 0
ò.ñ 0ò 0
ò.ñ 5ó 1.ñ 0ò0
ò3ô 0ò6
õ0ò9
ö0ò120ò15
ó0ò1÷ 8ø 0ò
Data grid andimage of original modelwith noise. ù denotesthepixel number,útheview direction.
InverseProblemsin SpacePhysicsII/10 May 2002
SVD-inversion: 2D tomography, the kernel
Tû
oü mý oü gþ rÿ a� p� h�y� mý a� t� rÿ i� x� K
�
moü d� e l s p� a� c� e (� , )�
d� a� t� a� s� p
� a� c� e� (� r,
� )�
Tomographykernelmatrix. Each subblock shows� for fixed á andú. Zero elementsareblank.
Ke� rne� l a� nd� d� a� t� a� s� p� e� c� t� ru m
1! 2" 1! 4# 1! 6$ 1! 8% 1! 1! 0& 1! 1! 2" 1! 1! 4# 1! 1! 6$ 1! 1! 8% 1!m' o( d� e� n) u m' b
*e� r+
-7,-6$-5--4
-3.-2"-1!0&1
lo
/ g0 1 a
1 nd
2 lo/ g0
1
Spectrumof kernel(crosses)anddata(circles)
InverseProblemsin SpacePhysicsII/11 May 2002
SVD-inversion: 2D tomography, the eigenfunctions
m3 o4 d5 e6 n7 o4 18 m3 o4 d5 e6 n7 o4 59
mo4 d5 e6 no4 20: mo4 d5 e6 no4 59 0:
m3 o4 d5 e6 n7 o4 18 0: 0: m3 o4 d5 e6 n7 o4 18 9; 2<
Eigenfunctions=?> of somemodesof the2D tomographykernel
InverseProblemsin SpacePhysicsII/12 May 2002
SVD-inversion: 2D tomography, reconstructions
t@ ruA ncB1
= 18 .C 0: ED -2< 0: 18 9; 2< ED VE sF aG cH cH pI 0: EVE sF re6 jJ cH t@ ruA ncB
1= 2< .C 8K ED -0: 6L 18 8K 6L ED VE sF aG cH cH pI
6L EVE sF re6 jJ cH
t@ rM uA nN cB1
= 1.C 1E-0: 4 18K 2EVE sF aG cH cH pI 18 0: ED VE sF rO e6 jJ cH t@ rM uA nN cB
1= 4.C 2E-0: 3P 16L 6L EVE sF aG cH cH pI
2< 6L ED VE sF rO e6 jJ cH
t@ ruA ncB1
= 18 .C 6L ED -0: 18 18 0: 4Q ED VE sF aG cH cH pI 8K 8K EVE sF re6 jJ cH t@ ruA ncB
1= 4Q .C 0: ED -0: 18 18 2< ED VE sF aG cH cH pI
18K 0: EVE sF re6 jJ cH
Reconstructionsfor varioustruncationlevels ÓSR cUTp]WVYX Ó InverseProblemsin SpacePhysicsII/13 May 2002
SVD-inversion: Generalizedinverses
For mostproblemsexactinversesE � i doesnotexist. �B
Theconceptof matrix inversesneedsto begeneralized.Gen-eralizedinverses
E �{iZ XJ: aredefinedthroughthefour Moore-Penrosecriteriafor generalizedinverses:
Insteadof beinga unit matrix,E �{iZ XJ: E and
E E �{iZ XJ: areonly re-quiredto besymmetric
�wE � iZ XJ: E � M � E �{iZ XJ: E (modelresolutionkernel)�wE E �{iZ X�: � M � E E � iZ XJ: (dataresolutionmatrix)
andthatthey actasunit matrixat leastin the“visible” modelsub-spacev �wE � i\[ G I andtherangez �wE � [ G L , respectively,E E � iZ XJ: E � EE � iZ XJ: E E � iZ XJ: � E �{iZ X�:We find that
E E �{i|~}�� satisfiesthesecriteriahowever its truncatedversion E �{i] |~}�� � ! R cUTp]WV
!hgji P ! �Q ! N !with U ë�ìíî : ï ¸ U :WVYX ì[Z satisfiesonly thefirst two Moore-Penrosecrite-ria, becauseE �{i] |®}A� E � ! R c^T`]WV
! gOi P ! P ! ' E E �{i] |~}�� � ! R cUTp]WV!hgji N ! N !
areprojectionoperatorsontoonly partof v �wE � iand z �wE �
, re-spectively.
InverseProblemsin SpacePhysicsII/14 May 2002
BG or mollifier inversion: Moti vation
Assumewehaveacontinuousmodelandadiscretenumberof ob-servations,i.e.,
� - Hilbert spaceandD - G L . For eachindividual
measurementU = 1, mnmom ,T wehave
� ! � � ! �������m�������$�������Theproblemis hoplesslyunderdeterminedanda conventionalin-verseof
E �����canneverbeachieved.
�BWe only want to obtainanestimateof
�������which shouldbe
a moreor lesslocalizedaverage.Sincetheproblemis linear thisestimatemustbea linearcombinationof thedata.For each
�find
coefficients _ ����� with
�¯������� L! gji ` ! �����&� ! � L
!hgji ` ! ����� � ! �������< =?> @�������$���"���
modelresolutionkernelX ���O'H�����
Theresolutionkernel(compareto Moore-Penrosedefinition)hereis a suitablelinearsuperpositionof theindividual forwardkernels� ! .Themodelresolutionkernel
X ���O'H� � �shouldsatisfyq Localizationwithin width a
Xcbedgf ' fihkjmlon õfor p fql fih p k ar b dgftsuf h jmlvlwnbyx{z | d}f~l f h jq Normalization rcbedgfts�fihkjw��fih�� �
InverseProblemsin SpacePhysicsII/15 May 2002
BG or mollifier inversion: SVD and noise
Assumewewereableto constructaSVD of thekernelfunctions
� d}foj�� �U�������^������
������?�
d}foj�s � � �
thenthe � ��g� j
span� � � j �completelyandthe resolutionkernelr �g� s �i¡ j
hasa representationin this basis.For simplicity we con-struct
r �}� s � ¡ jsothatit is diagonal:
r �g� s � ¡ j�� �U�������^������ � �
�g� j£¢� � �
�g� ¡ j
thentheequivalentinverseis
� ¤ �¥§¦©¨ª¨ �g� j«� �¬�W�����^��k�i� � �
�g� j ¢ ����� with
� ¤ �¥o¦£¨ª¨ �g� j� � r �g� s � ¡ jlon
no truncationasin TSVD but gentleroll-off dueto filter co-efficients
¢� .
If theobservations ® � arecontaminatedwith noise ¯ � thentheesti-mate° becomesaffectedaswell:
° �}� j§± °y² �g� j³� �U�������U��k�i�
´ ��g� j ® �
± �U�������U��k�i�
´ ��g� j ¯ �
If the thenoisehaszeromeanandvarianceµ·¶² theerror °�² of theestimatehaszeromeanandvarianceµ ¶² p¹¸ºp ¶ . l»n
To confinetheerrordueto datanoiseweneedasadditionalrequirement:¼ Shortestpossible
�U�������^����i�
´ ¶��}� j½n
minimum
InverseProblemsin SpacePhysicsII/16 May 2002
BG or mollifier inversion: Mollification
For each�
try to find coefficients ´ ��g� j
so thatrc¾ �g� s �i¡ j
comescloseto a desiredmollifier function | ¾ �}� s � ¡ j with width ¿ , i.e.,solve (usuallyby SVD)À
�k�i�´ ��}� jÂÁ
��g� ¡ j�� | ¾ �g� s � ¡ jo± ÃÄÆÅ �g� ¡ j
whereÃÄÆÅ �g� ¡ j�Ç � � � j
is thepartof themollifier which falls intothenullspaceof
�. Tunewidth ¿ sothattheerror È p¹¸ºp doesnot
exceedgivenbounds.
Disadvantages:¼ The above equationhasto be solved for every�
at which anestimate° is required.Note,however, that the above equationismucheasierto solve thantheoriginalproblembecausethereis nonoiseinvolved.¼ The computationaloverheadis large unlesssymmetriesof thesystemreducethe numberof resolutionkernels
rc¾ �}� s � ¡ jto be
calculated.
Advantages:¼ For every�
we not only obtainan estimate° of the modelbutalsoa resolutionkernel
r ¾ �g� s � ¡ jtelling us which region ° �g� j is
representativeof. Wealsoobtainanindividualerrorestimateµ ² p¹¸ºpfor each° .¼ Thereis noneedto discretizethemodelspace¼ Theresolutionkernels
rc¾ �}� s � ¡ jcanbeusedagainwith different
dataif thekernelsÁ��g�i¡ j
havenot changed
InverseProblemsin SpacePhysicsII/17 May 2002
BG or mollifier inversion: Gaussianmollifier
20É
40É
6Ê 0É 8Ë 0É 10É
0É0
É.Ì 0É0É
.Ì 2Í0É
.Ì 4Î0É
.Ì 6Ê0É
.Ì 8Ë1Ï .Ì 0É
keÐ rneÐ l
fu
Ñ ncÒ tÓ io
Ô nhÕeÖ i× gØ hÕ tÙ
keÖ rneÖ l K iÚ (Û x)Ü
rÝ eÖ sÞ oß là uá tÙ i× oß nâ kã eÖ rÝ nâ eÖ là aä nâ då mæ oß là là i× fç i× eÖ rÝ
0è
.é 0è 0è 0è0è
.é 0è 0è 5ê0è
.é 0è 10è0
è.é 0è 1ë 5ê
0è
.é 0è 2ì 0èkã eÖ rÝ nâ eÖ là cí oß eÖ fç fç i× cí i× eÖ nâ tÙ sÞ iÚ
-0è
.é 0è 0è 2-0è
.é 0è 0è 1ë0è
.é 0è 0è 0è0è
.é 0è 0è 1ë0è
.é 0è 0è 2ì
0è
.é 0è 0è0è
.é 0è 1ë0è
.é 0è 2ì0è
.é 0è 3î0è
.é 0è 4ï
-0è
.é 0è 10è
.é 0è 0è0è
.é 0è 1
hÕeÖ i× gØ hÕ tÙ20
è40è
6ð
0è
8ñ 0è 10è
0è
0è
.é 0è 0è0è
.é 0è 5ê0è
.é 1ë 0è0è
.é 15ê
0è
.é 2ì 0è
kã eÖ rÝ nâ eÖ là nâ uá mæ bò eÖ rÝ i×20è
40è
6ð
0è
8ñ 0è 10è
0è-10
è-5ê0è5ê
1ë 0è
Mollifiers ó andresolutionkernelsô·õ (left) andkernelcoefficientsö ÷ (right) forthekernels ø�÷ in thetopdiagram.Resolutionkernelsarederivedfor ù (height)= 55 and different width ú . Resolutionkernelsand mollifiers are practicallyidentical.
InverseProblemsin SpacePhysicsII/18 May 2002
BG or mollifier inversion: Box-shapemollifier
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Mollifiers ó andresolutionkernelsô·õ (left) andkernelcoefficientsö ÷ (right) forthekernels ø�÷ in thetopdiagram.Resolutionkernelsarederivedfor ù (height)= 55 anddifferentwidth ú . Mollifiersareexactlybox-shape.
InverseProblemsin SpacePhysicsII/19 May 2002
BG or mollifier inversionNoisecomparison
Theamountof noisein theestimate° �}�oü is µ ²þý¹ÿºý . Here, µ ² is thestandartdeviation of thenoisein thedata,and ý¹ÿ½ý is the lengthofthekernelcoefficient vectorgivenbelow.
0�
2 4 6� 8� 10�
r� e� s� o� l� u� t i o� n� k� e� r� n� e� l� w id�
th�-4
�-2�0�2
4
lo
� g� le� ng� t� h |� |�
G� a� u� s� s� i a� n� m� o� l� l� i f� i e� r�
b�o� x� s� h� a� p� e� m� o� l� l� i f� i e� r�
Lengthof � vs width w of the resolutionkernel for a Gaussianandbox-shapemollifier
InverseProblemsin SpacePhysicsII/20 May 2002
BG or mollifier inversion: Backus-Gilbert approach
We do not specifythe shapeof the resolutionkernelbut only tryto concentrateits width arounda given
�by minimizing
�g� � � ¡ ü ¶"! ¶ �g�$#�� ¡ ü&%�� ¡(' �g� � � ¡ ü ¶*) À
�����´ ��g�oü Á
��g� ¡ ü,+ ¶ %��
' À
�.- /Â�i� ´ ��}�oü ´ / �g�oü �g�0� � ¡ ü ¶ Á �
�g� ¡ ü Á / �g� ¡ ü&%��21 � ÿ �g�oü4365 �g�oü ÿ �g�oüü
This expressionhasto be minimizedalongwith µ·¶² ý¹ÿ �g�oü ý ¶ (noisereduction)underthenormalizationconstraint7 ' ! �g�$#�� ¡ ü&%�� ¡ ' À
�k�i�´ ��g�oü Á
��g� ¡ ü&%��21 � ÿ �g�oü4398 ü
UsingLagrangianmultipliers : and ; , thecoefficient vector ÿ �g�oüis determinedby
� ÿ 3<5 ÿ ü ± :tµ ¶² � ÿ 3 ÿ ü ± ;>= � ÿ 398 ü?� 7"@ �BAminimum
for known matrix5
andvector8
.
Theresultis ÿ ' 7�C8D3 = 5 ± :tµ ¶²FE @ ¤ � 8 ü = 5 ± :tµ ¶² E @ ¤ � 8
which hasto be solved for every�. The parameter: serves to
balanceresolutionvsnoiseandstabilizetheinversionof the�2G �
matrix5 ± :tµ ¶² E .
¼ Theresulting! �g�$#�� ¡ üis well concentratedaround
�but yetmay
not be well centeredon�. Therefore,an additionalconstraintis
sometimesusedto obtainwell centeredresolutionkernels
InverseProblemsin SpacePhysicsII/21 May 2002
BG or mollifier inversion: Tomography
In tomography the index H standsfor pixel number¢
and viewdirection I . In 2D:
®KJML -ONQP ' Á JRL -SNTP �g� ¡ ü ° �g� ¡ ü&%�� ¡ # where� Ç U ¶ # ¢ ÇVU
andÁ JRL -ONQP �g�i¡�ü is thebeamfrom pixel
¢into direction W NÁ JML -SNTP �g� ¡ ü ' 7
if� ¡
insidethebeam(¢ # I )X
else
Themollifier methodseeks ° �g�oü ' L -ON´ JRL -ONQP �g�oü ® JML -SNTP with
´ JML -ONQP sothat ! �}�$#�� ¡ ü ' L -ON´ JRL -SNTP �g�oü Á JRL -ONQP �g� ¡ ü �BA Y��g� � � ¡ ü
hence, for each�
find co-efficients ´ JML -SNTP �}�oü so thattheresultingsuperpositionofbeamsapproachesa
Yfunc-
tion at�.
In filteredbackprojectiontomography thespecialchoiceis´ JRL -ONQP �g�oü ' ¿ L ¤ L[Z Á JRL\Z -ONQP �g�oü where
¢^]sothat
Á JRL\Z -ONQP �g�oü`_' XThis givesa symmetricresolutionkernel ! �g�$#u�i¡�ü
and
° �g�oü ' NÁ JML[Z -SNTP �g�oüa bdc e L ¿ L ¤ L\Z ® JML -SNTPa bdc ebackproj filter
InverseProblemsin SpacePhysicsII/22 May 2002
Conclusions:What is the problemwithinverseproblems?
Kernelfunctionsare“smooth” in thesense(Riemann-Lebesque)
Á �g�$#�� ¡ ügfihkj Åml �i¡Åonqprl � ¡ts %�� ¡ �uA Xas
l �uA v�wA ® insensitive to theshortwavelengthstructurein °�wA
solvingfor ° is anill-posedproblem(Hamadard):¼ ° is eithernotunique(nullspaces)¼ ° changesdiscontinuouslywith ® (smalleigenvaluesofÁ
)
Whatis thesolutionto theproblemwith inverseproblems?Replacetheorginalproblemby aseriesof solvableproblems:
® �}�oü ' Áyx �g�$#u� ¡ ü ° x �g� ¡ ü&%�� ¡ with z nC{xk|~} Áyx �g�$#u� ¡ ü ' Á �g�$#u� ¡ üandset ° ' z nC{xk|~} ° x . Examplesfor theregularizationparameter:� ' 7^� l������t�t� in FT inverstion' 7^� H �����t�t� in SVD inverstion' � nC�K�o� ¿ in mollification' : in Backus-Gilbertinversion
In practicalcases,however, we have to stopat a finite � duetonoise.Thekey problemsare:¼ to find theoptimumvalue ��� of � ,¼ to understandwhich featuresof ° xk� will survive if wecould
let � �wA 0¼ which featuresz nC{xk|~} ° x might havewhich ° x�� doesnothave.¼ which contribution from � � Á ü
hasto beaddedto ° xk� .InverseProblemsin SpacePhysicsII/23 May 2002