CS 395/495-26: Spring 2004
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CS 395/495-26: Spring 2004CS 395/495-26: Spring 2004
IBMR:IBMR:Projective 2D DLTProjective 2D DLT
Jack TumblinJack Tumblin
[email protected]@cs.northwestern.edu
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x’x’
Apply the Apply the 3x3 matrix 3x3 matrix HH
x’ = Hx x’ = Hx
Recall: Projective Transform Recall: Projective Transform HH
xx33
xx22
xx11
xx
yy
x’x’
y’y’
(x(x/w/w,y,y/w/w)) xx
(x’(x’/w’/w’,y’,y’/w’/w’))
Homog. coordsHomog. coords x’ =x’ = [x’,y’,[x’,y’,w’w’]]TT
w’=1w’=1
2D image 2D image (x’,y’)(x’,y’)
2D image 2D image (x,y)(x,y)
Homog. coords Homog. coords
[x,y,[x,y,ww]]TT = x = x w=1w=1
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Recall: Projective Transform: Recall: Projective Transform: Hx = x’Hx = x’
xx11
xx22
xx33
hh11 11 h h1212 h h1313
hh2121 h h2222 h h2121
hh3131 h h32 32 h h3333
= = x’x’11
x’x’22
x’x’33
H defines point pairs (x,x’) H defines point pairs (x,x’) (correspondences)(correspondences)
x’ = =x’ = =
y’ = =y’ = =
= = xx
yy
ww
Input (or output) Input (or output) image is on image is on central planecentral plane
xx11
xx22
xx33
hh21 21 x + hx + h22 22 y + hy + h2121
hh31 31 x + hx + h32 32 y + hy + h3333
x’x’11
x’x’33
x’x’22
x’x’33
Write RWrite R22 expressions; expressions;(for each x,x’ pair)(for each x,x’ pair)
hh11 11 x + hx + h12 12 y + hy + h1313
hh31 31 x + hx + h32 32 y + hy + h3333
x’ (hx’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))
y’ (hy’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))Rearrange, solve as Rearrange, solve as a matrix problem…a matrix problem…
x = [x = [xx11, x, x22, x, x33] notation ] notation
is confusing for multiple points!is confusing for multiple points!
Instead, let’s use [x,y,w]:Instead, let’s use [x,y,w]:(so we can use subscripts(so we can use subscripts to number the point pairs x,x’…)to number the point pairs x,x’…)
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Recall: Projective Transform: Recall: Projective Transform: Hx = x’Hx = x’
xx11
xx22
xx33
hh11 11 h h1212 h h1313
hh2121 h h2222 h h2121
hh3131 h h32 32 h h3333
= = x’x’11
x’x’22
x’x’33
H defines point pairs (x,x’) H defines point pairs (x,x’) (correspondences)(correspondences)
x’ = =x’ = =
y’ = =y’ = =
= = xx
yy
ww
Input (or output) Input (or output) image is on image is on central planecentral plane
xx11
xx22
xx33
hh21 21 x + hx + h22 22 y + hy + h2121
hh31 31 x + hx + h32 32 y + hy + h3333
x’x’11
x’x’33
x’x’22
x’x’33
Write RWrite R22 expressions; expressions;(for each x,x’ pair)(for each x,x’ pair)
hh11 11 x + hx + h12 12 y + hy + h1313
hh31 31 x + hx + h32 32 y + hy + h3333
x’ (hx’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))
y’ (hy’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))Rearrange, solve as Rearrange, solve as a matrix problem…a matrix problem…
let w=1let w=1
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Recall: Projective Transform: Recall: Projective Transform: Hx = x’Hx = x’
xx11
xx22
xx33
hh11 11 h h1212 h h1313
hh2121 h h2222 h h2121
hh3131 h h32 32 h h3333
= = x’x’11
x’x’22
x’x’33
Finding H from point pairs Finding H from point pairs (correspondences)(correspondences)
x’ = =x’ = =
y’ = =y’ = =
= = xx
yy
1 1
Input (or output) Input (or output) image is on image is on central planecentral plane
xx11
xx22
xx33
hh21 21 x + hx + h22 22 y + hy + h2121
hh31 31 x + hx + h32 32 y + hy + h3333
x’x’11
x’x’33
x’x’22
x’x’33
Write RWrite R22 expressions; expressions;(for each x,x’ pair)(for each x,x’ pair)
hh11 11 x + hx + h12 12 y + hy + h1313
hh31 31 x + hx + h32 32 y + hy + h3333
x’ (hx’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))
y’ (hy’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))Rearrange, solve as Rearrange, solve as a matrix problem…a matrix problem…
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Naïve Method:Naïve Method:
4 point correspondence:4 point correspondence:Book shows:Book shows:
Rearrange: Rearrange: knownknown vector (dot) vector (dot) unknownunknown vector vector
stack, solve for null space…stack, solve for null space…
But this assumes But this assumes xx11=x, x=x, x22=y, =y, xx33=w=1=w=1,,
There is a better way (DLT)…There is a better way (DLT)…
x’ (hx’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))
y’ (hy’ (h31 31 x + hx + h32 32 y + hy + h3333) = (h) = (h11 11 x + hx + h12 12 y + hy + h1313))
hh1111
hh1212
hh13 13
hh2121 hh2222
hh2323
hh3131 hh3232
hh3333
[x y 1 0 0 0 -x’x -x’y -x’] = 0[x y 1 0 0 0 -x’x -x’y -x’] = 0[0 0 0 x y 1 -y’x -y’y -y’][0 0 0 x y 1 -y’x -y’y -y’]
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Direct Linear Tranform (DLT)Direct Linear Tranform (DLT)
• Naive method: Solve Hx – x’ = 0Naive method: Solve Hx – x’ = 0– Requires constant scale (xRequires constant scale (x33=1 always)=1 always)
• compare:compare:
• A BETTER WAY:A BETTER WAY:DLT method: Solve Hx DLT method: Solve Hx x’ = 0x’ = 0– Accepts any scale (xAccepts any scale (x33=w=w1 OK…)1 OK…)
[ 0 0 0 x y 1 -y’x -y’y -y’ ] [ 0 0 0 x y 1 -y’x -y’y -y’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ]
hh1111
hh1212
hh1313
hh2121
hh2222
hh2323
hh3131
hh3232
hh3333
[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[w’x w’y w’w 0 0 0 -x’x -x’y -x’w] [w’x w’y w’w 0 0 0 -x’x -x’y -x’w]
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Direct Linear Tranform (DLT)Direct Linear Tranform (DLT)
• Naive method: Solve Hx – x’ = 0Naive method: Solve Hx – x’ = 0– Requires constant scale (xRequires constant scale (x33=1 always)=1 always)
• Compare to DLT:Compare to DLT:
• DLT method: Solve Hx DLT method: Solve Hx x’ = 0x’ = 0– Accepts Accepts anyany scale (x scale (x33=w=w1, x1, x33’=w’’=w’1)1)
[ 0 0 0 x y 1 -y’x -y’y -y’ ] [ 0 0 0 x y 1 -y’x -y’y -y’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ][ x y 1 0 0 0 -x’x -x’y -x’ ]
hh11
hh22
hh33
[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[w’x w’y w’w 0 0 0 -x’x -x’y -x’w][w’x w’y w’w 0 0 0 -x’x -x’y -x’w]
[[ 00TT --ww’’i i xxiiTT yy’’I I xxii
TT ]][[ ww’’i i xxii
TT 00TT - -xx’’i i xxiiTT ]]
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Direct Linear Tranform (DLT)Direct Linear Tranform (DLT)
• DLT method: Solve H DLT method: Solve H x = 0x = 0– Accepts any scale, any point (xAccepts any scale, any point (x33=w=w1 is OK)1 is OK)
– ‘‘Pure’, Compatible –uses only Pure’, Compatible –uses only PP22 terms.terms.– Much better suited to error measurements.Much better suited to error measurements.
• Subtlety:Subtlety:– Won’t divide-by-zero if Won’t divide-by-zero if w=0w=0 or or hh3333=0=0 (it happens!)(it happens!)
– has a 3has a 3rdrd row; it is not degenerate if/when w’=0 row; it is not degenerate if/when w’=0– OK to use it…OK to use it…
(Solve 8x12)(Solve 8x12) hh11
hh22
hh33
00TT - -ww’’i i xxiiTT yy’’I I xxii
TT ww’’i i xxii
TT 00TT - -xx’’i i xxiiTT =0 =0
yy’’i i xxiiTT - -xx’’i i xxii
TT 00TT
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STOP:STOP:Learn about SVDs …Learn about SVDs …
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Matrix Math & SVDsMatrix Math & SVDs
• Matrix multiply is a change of coordinatesMatrix multiply is a change of coordinates– Rows of A = coordinate axis vectorsRows of A = coordinate axis vectors– Ax = a dot product for each axisAx = a dot product for each axis
• input spaceinput spaceoutput spaceoutput space• SVD ‘factors’ matrix A into 3 parts:SVD ‘factors’ matrix A into 3 parts:
– U columns: orthonormal axes for OUTPUTU columns: orthonormal axes for OUTPUT– V columns: orthonormal axes for INPUTV columns: orthonormal axes for INPUT– S: scale factors VS: scale factors VUU
svd(A) = U S Vsvd(A) = U S VTT
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Actual ‘Robustness’Actual ‘Robustness’
• Vectorizing Vectorizing (‘Flatten, Stack, Null Space’ method) (‘Flatten, Stack, Null Space’ method)
works for almost ANY input! works for almost ANY input! (Points, lines, planes, ||, (Points, lines, planes, ||, , conics, quadrics, , conics, quadrics, cross-ratios, vanishing points, twisted cubics…cross-ratios, vanishing points, twisted cubics…
• Key to DLT formulation: (Hx Key to DLT formulation: (Hx x’) = 0 x’) = 0
• Rearrange as dot product: Rearrange as dot product: (known)(known)··(unknown) = 0(unknown) = 0
• Be careful to have ENOUGH constraintsBe careful to have ENOUGH constraints tricky when you mix types: points, lines,… tricky when you mix types: points, lines,… (pg 75)(pg 75)
measured inputmeasured inputtransformedtransformed
measured outputmeasured output
= 0= 0
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Deceptions: false ‘Robustness’…Deceptions: false ‘Robustness’…
• Suppose we have 4 pt-correspondences:Suppose we have 4 pt-correspondences:– Use DLT to write 8x9 (or 12x9) matrix Use DLT to write 8x9 (or 12x9) matrix AA: : Ah=0Ah=0– Solve for Solve for h h null space. null space. ALWAYSALWAYS gives gives HH matrix matrix
• But what if points are bad / fictional?But what if points are bad / fictional?– 3 collinear input pts, crooked out: 3 collinear input pts, crooked out: IMPOSSIBLE!IMPOSSIBLE!
– Yet we get an Yet we get an HH solution! Why? solution! Why?
• AA matrix rank is 7 or 6 matrix rank is 7 or 6 rank 1 rank 1 HH result(s) result(s)– ‘‘Null Space’ of Null Space’ of AA may contain >1 may contain >1 HH solution! solution!
– ‘‘Degenerate’ Degenerate’ HH solution of form solution of form aaHH11 + + bbHH22……
• Answer: SVD ranks Answer: SVD ranks AA; ; rejectreject bad point sets. bad point sets.
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Most Important SVD IdeasMost Important SVD Ideas
SVD solves Null-Space problems SVD solves Null-Space problems Ax = 0Ax = 0 (‘flatten, (‘flatten,
stack, find null space’ to solve) stack, find null space’ to solve) – MANY problems can be written this wayMANY problems can be written this way– Expandable! Expandable!
--more data points--more data points tall, ‘overconstrained’ A matrix tall, ‘overconstrained’ A matrix --SVD finds optimal ‘least squares’ solution.--SVD finds optimal ‘least squares’ solution.
Robustness: bane of all inverse methodsRobustness: bane of all inverse methodsAccuracy vs. # of measurements Accuracy vs. # of measurements (more isn’t always better! Outliers...)(more isn’t always better! Outliers...)
Data-rich images: Data-rich images: quantityquantity easier than easier than qualityquality moremore is easier than is easier than betterbetter) )
?What links measurement errors?What links measurement errorsHH errors? errors??How can more measurements reduce error??How can more measurements reduce error?
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HowHow else else can we use DLT? can we use DLT?
• I know this building I know this building consists of MANY parallel consists of MANY parallel and perpendicular lines, and perpendicular lines, corners, regularly spaced corners, regularly spaced points.points.
• How many How many differentdifferent ways ways can we find to ‘rectify’ this can we find to ‘rectify’ this image to get this one?image to get this one?
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Direct Linear Tranform (DLT)Direct Linear Tranform (DLT)
• Naïve method: Solves Hx – x’ = 0Naïve method: Solves Hx – x’ = 0– Uses constant scale (xUses constant scale (x33=w=1 in measurement)=w=1 in measurement)
• Compare to DLT:Compare to DLT:
• DLT method: Solve Hx DLT method: Solve Hx x’ = 0x’ = 0– (a different identity: x’ (a different identity: x’ x’ = 0 x’ = 0))
– Accepts any scale (xAccepts any scale (x33=w=w1 OK, xx1 OK, xx33))
[ 0 0 0 x y 1 -y’x -y’y -y’ ] [ 0 0 0 x y 1 -y’x -y’y -y’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ]
hh1111
hh1212
hh1313
hh2121
hh2222
hh2323
hh3131
hh3232
hh3333
[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[w’x w’y w’w 0 0 0 -x’x -x’y -x’w] [w’x w’y w’w 0 0 0 -x’x -x’y -x’w]
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Direct Linear Tranform (DLT)Direct Linear Tranform (DLT)
• Naïve method: Solve Hx – x’ = 0Naïve method: Solve Hx – x’ = 0– Requires constant scale (xRequires constant scale (x33=1 always)=1 always)
• compare:compare:
• DLT method: Solve Hx DLT method: Solve Hx x’ = 0x’ = 0– Accepts Accepts anyany scale (x scale (x33=w=w1, x1, x33’=w’’=w’1)1)
[ 0 0 0 x y 1 -y’x -y’y -y’ ] [ 0 0 0 x y 1 -y’x -y’y -y’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ] [ x y 1 0 0 0 -x’x -x’y -x’ ]
hh11
hh22
hh33
[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[ 0 0 0 -w’x -w’y -w’w y’x y’y y’w] = 0[w’x w’y w’w 0 0 0 -x’x -x’y -x’w][w’x w’y w’w 0 0 0 -x’x -x’y -x’w]
[[ 00TT --ww’’i i xxiiTT yy’’I I xxii
TT ]][[ ww’’i i xxii
TT 00TT - -xx’’i i xxiiTT ]]
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Direct Linear Tranform (DLT)Direct Linear Tranform (DLT)
• DLT method: Solve H DLT method: Solve H x = 0x = 0– Accepts any scale, any point (xAccepts any scale, any point (x33=w=w1 is OK)1 is OK)
– ‘‘Pure’ and Compatible -- Pure’ and Compatible -- PP22 terms only, no Rterms only, no R22
– MuchMuch better suited to error measurements. better suited to error measurements.
• Subtlety:Subtlety:– Won’t divide-by-zero if Won’t divide-by-zero if w=0w=0 or or hh3333=0=0 (it happens!)(it happens!)
– has a 3has a 3rdrd row; it is row; it is notnot degenerate if w’=0 degenerate if w’=0– OK to use it…OK to use it…
(Solve 8x12)(Solve 8x12) hh11
hh22
hh33
00TT - -ww’’i i xxiiTT yy’’I I xxii
TT ww’’i i xxii
TT 00TT - -xx’’i i xxiiTT =0 =0
--yy’’i i xxiiTT - -xx’’i i xxii
TT 00TT
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Adding More MeasurementsAdding More Measurements
How can we use How can we use >4>4 point correspondences? point correspondences?• Easy:Easy:
– Add more rows to our 8x9 matrix A: Add more rows to our 8x9 matrix A: AA··h = 0h = 0– Use SVD to find Null space Use SVD to find Null space (Always gives an answer!)(Always gives an answer!)
– Result:Result: ‘Least squares’ solution; error= ‘Least squares’ solution; error=AA··h = h = • minimizes || minimizes || A hA h || ||22 = = ii 22
i i
wherewhere,,ii is error for is error for ii-th pt. -th pt. correspondence: correspondence:
i i = || Hx= || Hxii x xii’ ||’ ||2 2 = = || (2 rows of A)*h |||| (2 rows of A)*h ||2 2 = ‘algebraic distance’= ‘algebraic distance’
– ‘‘Algebraic Distance’ ?Algebraic Distance’ ? No geometric meaning! No geometric meaning!
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Adding More MeasurementsAdding More Measurements
• 2D‘Algebraic Distance’ ?2D‘Algebraic Distance’ ? No geometric meaning! No geometric meaning!
• 2D ‘Geometric Distance’ d2D ‘Geometric Distance’ d(a,b)(a,b)22 is Better: is Better: measurable length in input or output spacemeasurable length in input or output space
if a = (aif a = (a11 a a22 a a33) and b = (b) and b = (b11 b b2 2 b b33), then define), then define
Turns out that:Turns out that:
( - )( - )22 + + ( - )( - )22 dd(a,b)(a,b)22 = = aa11
aa33
bb22
bb33
bb11
bb33
aa22
aa33
dd(a,b)(a,b)22 = d = dalgebraicalgebraic(a,b)(a,b)aa33··bb33 (Not very surprising)(Not very surprising)
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Adding More MeasurementsAdding More Measurements
Iterative Strategy:Iterative Strategy:• Overconstrain the answer Overconstrain the answer HH
– Collect extra measurements (>4 point pairs, etc. …)Collect extra measurements (>4 point pairs, etc. …)– expect errors; call them ‘estimates’ xexpect errors; call them ‘estimates’ x
• Compute a 1Compute a 1stst solution (probably by SVD) solution (probably by SVD)
• Compute error Compute error d(Hx, x’)d(Hx, x’)22, and use this to…, and use this to…
• ‘‘Tweak’ answer Tweak’ answer HH andand estimates estimates xx
• Compute new answerCompute new answer
• Stop when error < useful thresholdStop when error < useful threshold
^̂
^̂
^̂
^̂
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Using EstimatesUsing Estimates
• Simplest: ‘one image’ transfer method:Simplest: ‘one image’ transfer method:– Assume inputs are a perfect test pattern:Assume inputs are a perfect test pattern:
onlyonly output output pts are in error pts are in error– Adjust output estimates x’ until d(Adjust output estimates x’ until d(HHx, x’)x, x’)22 00
(note we re-compute H as x’ changes)(note we re-compute H as x’ changes)
• Better: ‘Symmetric’ transfer method:Better: ‘Symmetric’ transfer method:– Assume BOTH input Assume BOTH input xx and output and output x’x’ has error. has error.– Adjust BOTH input and output estimates Adjust BOTH input and output estimates x x’ x x’
(note we also re-compute (note we also re-compute HH as as x, x’x, x’ change) change)– Stop when d(Hx,x’)Stop when d(Hx,x’)22 + d(H + d(H-1-1x’, x)x’, x)22 is nearly 0 is nearly 0
^̂ ^̂
^̂ ^̂
^ ^ ^ ^ ^ ^ ^ ^
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