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± íÄÆÅ �g� ¡ j

whereíÄÆÅ �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