CPSC 4040/6040 Computer Graphics Images Joshua Levine - - PowerPoint PPT Presentation

CPSC 4040/6040 Computer Graphics Images

Joshua Levine levinej@clemson.edu

Lecture 20 Removing Warp Artifacts

• Nov. 5, 2015

Slide Credits: Szymon Rusinkiewicz

Agenda

Refresher from Lec19

Projective Warps

• What is the matrix?
• What the knowns? Unknowns? How many?

Algebra

• A general 3x3 matrix can express this entire class of warps (9 unknowns)
• a33 acts as a global scale parameter, so we can always set it to 1

without losing generality

• The remaining 8 unknowns can be solved by 4 pairs of equations using

the 8 known xi, yi values and the 8 known ui, vi values

• Solving these 8 equations gives the 8 remaining aij unknowns

Bilinear Warping

• Key Idea: Instead of

using bilinear interpolation for pixel color values, we can use it to interpolate the positions in the warped image

Step 2: Forward Warp with u,v offsets

Inverse Bilinear Warp Can Be Computed from the Forward Warp

• Forward warp for a pixel (s, t) is equivalent to the following equations:
• where (s0, t0) is the lower left of the image, (s1, t1) is the upper right of

the image, and (effectively normalizing)

Inverse Bilinear Warp Can Be Computed from the Forward Warp

• Taking the inverse of
• Leads to
• Where 0<u<1, 0<v<1, and

Recall: Converting Between Image Domains

• When an image is acquired,

an image is taken from some continuous domain to a discrete domain.

• Reconstruction converts

digital back to continuous through interpolation.

• The reconstructed image can

then be resampled and quantized back to the discrete domain.

Fixing Jaggies / Magnification Artifacts

Reconstruction Artifacts

• Leads to staircasing or “jaggies”

Do a Better Reconstruction?

• Basic Idea: If we interpolate the data samples

better we will have a superior reconstruction

• How? Bilinear Interpolation, Bicubic, etc.

Recall: Nearest Neighbor

Recall Bilinear Example

Recall Bicubic (from Photoshop)

Ignore small color issues

Fixing Aliasing / Minification Artifacts:

Aliasing Artifacts

• Aliasing leads

to missing and/

• r unwanted

features

• Example:

12x12 images scaled to a 4x4 image.

Aliasing

• When we minify, we use only a few samples to represent lots of data
• High frequencies “masquerade” as low ones
• Images look “ropey”. This is not jaggies!
• (Barely) adequate sampling

Inadequate sampling

Aliasing Described by Sampling Theory

• What happens if we use too few samples?
• Aliasing: when high frequencies masquerade as low ones

How many samples are enough to avoid aliasing?

How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?

How many samples are enough to avoid aliasing?

How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?

How many samples are enough to avoid aliasing?

How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?

How many samples are enough to avoid aliasing?

How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?

How many samples are enough to avoid aliasing?

How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?

Antialiasing Filters

• Basic Idea: Smooth the image first to reduce the
• verall frequency
• How? Use filters!

Resampling with Filters

• Output is weighted average of inputs:

float Resample(src, u, v, k, w) { float dst = 0; float ksum = 0; int ulo = u - w; etc. for (int iu = ulo; iu < uhi; iu++) { for (int iv = vlo; iv < vhi; iv++) { dst += k(u,v,iu,iv,w) * src(u,v) ksum += k(u,v,iu,iv,w); } } return dst / ksum; } Source image Destination image f (u,v) (ix,iy)

Local Convolution?

• Compute weighted sum of pixel neighborhood

Output is weighted average of input, where weights are normalized values of filter kernel (k)

(u,v)

k(ix,iy) represented by gray value

dst(ix,iy) = 0; for (ix = u-w; ix <= u+w; ix++) for (iy = v-w; iy <= v+w; iy++) d = dist (ix,iy) dst(ix,iy) += k(ix,iy)*src(ix,iy);

w (ix,iy) d

Smoothing an Image in This Way Limits the Frequency Bands

Point Sampled: Aliasing! Correctly Bandlimited

Another Approach

• Instead of

smoothing, we can also sample better and then aggregate the samples

Artifact “Free” Warping Pipeline

• Ideal resampling

requires correct filtering to avoid artifacts

• Reconstruction filter

especially important when magnifying

• Bandlimiting filter

especially important when minifying

Sample Real world Reconstruct Discrete samples (pixels) Transform Reconstructed function Filter Transformed function Sample Bandlimited function Reconstruct Discrete samples (pixels) Display

Warping Pipeline

Lec21 Required Reading

• House, Ch. 13

Feature-Based Image Metamorphosis

Comtruter GraDhics, 262, Julv 1992 7’llcI1ldljll\ Bcicr Silicon Graphics C’(mlpulcr Systc]ms 201 I Shorclirm Blvd. Moun(ain View CA 94043

/v(’(’/v Pxi tic Ekild lnlagc~ I 1I 1 Karlstxi Drive. Sunny\alc CA 94)X9

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Abstract Kc>u cwds: (’tnnpulcr Aninwi(m. Interpolation. lnwgc f%~ccsilng. Sh;ipc ‘1’r~illit(~rlll:ilit)ll.

2

Introduction 2.1 Conventional Metamorphosis Techniques Mc[:ml(wpht)iii twlween lWo or mor’c imafys (wer lime i) u uwi’ul \ i~u;ii tcchniquc. (Jflen uwd f’orCducaliomd (n’tMCid;liMll Cnt pur- pt>wi. ‘1’l-:idi(ional Iilmmahing techniques for (his cflcc[ include ~’lckcrc’ut~(iuc’h LIS u chwwwr cxhibi(ing ch:mgm while running thr(mgll ;! toreil and prosing behind several trws ) tind op[ic:d cro\\- diswdv<’. in which onc image is f:ide(i out while wwther is sinwlt:l- nLNNI\l)f’:idcdin (Mith makeup ch:mge. tippliwcm,

• r nhjecl subs[i -

[u[I(m ). Sc\’~’riilclawic horror lilm~ illu$tfiite [he process: who ctwld

hnycl ~hcb:lir-tai~ing (fiiniform;ilml

• f the Woitman. or the drw

m:itic lllct;itll(~rpll(~sii from Dr. Jchyll [o Mr. Hyde’? This pupcr prcwmls ii c(mtcnlp{mmy w~lu(i(mto the vi~u:d translonmrtion pnh lL’nl.

‘fiIh In: the cutIIng

appro:~~h to Ihc limit giws us the techniqw 01” il(q>-nl~xi(m :minmtion. in which the subject is progres~ively tran\- I’[mncd mrd ph(~togr:tphed tme fr:mw at a ;imc. This process c:m give the Ixmcrl’ul illusi(m of cmltinu(ms rnetamwphosis. but it require~ much skill and IS\cr! tedi~ms worh. Moreover. stop-motion tr~uully wfl”cm t’r(mlthe prt~hlcm ()( \ iiu;il itrobing by not prm iding the nl~~li~m blur n(mn:lll! :i~w}ciatcd wi(h rowing film suh,jecls. A m(~- lmn-c~mlrt)lled variwrt ctillcd gmm{)ti{m (In which the frame-h) f’rmnc \uhjccl) art! photogrtiphcd while mwlrrg) can provide [he pr{)permotion Murtocretite timorc ntitwd effect. but the cornplcxlty (i the m(deli. moti(m hfirdww. wrd required skill$ hecnmc~ mm ~,rcaw. > 2.2 3D Computer Graphics Techniques We ctin uw technology in other V.U!SI(}help build u rnc[amorphoii~ 1(x)1.For cxwnplc, w can usc computer gruphic~ to rnodcl and rcndcl- lm;igcs which trim~fornl ()~cr time.

• nc ~ippr(xich in~(~lwi the reprcwntatlon
• f a pair ot threc-dtmen-

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[~biects ;ISa collecti(m (}Ipol}gon~. The vcrticw of the first

• h,jec( :Ir; then d]splwwd (wr

time to coincldc in po~ition u Ith ~x~rrcsponding icrtice~ of the wumd (hject, v.ith color and olhcr :ittrihutes similwl~ irwrpnltitcd. The chief prohlern wlth (his [cuh- n]quc ii the difficulty in eittiblishing a de~lrahle YWCAc(m_c\fx)n-

dcncc: thii ~)f’tcn

impow inconvenient cmrstrainti on the gw]mclric rcprcwntu{inn of the objects, wch ;is requiring the wrne number of pt~l}g(nli in c;lch model. Even rf thtw c(mditions tire met. problems ilill :mw when the (npologief ~)lthc two objects Lliffcr{\uch tI\ when [me (~bject hiis ii hole thrnugh it ),{mwhen the features mu~t M(W in a comple~ vii! (such m ~iiding al(mg [he object wrf;w lrom back I()t’r(mt). This direct point-lntcrp{)ltitlon” technique can be effective, ht~wckcr. for transformations in which the data corre~pondcncc and lntcrpol; ition p;lths are slmplc. For cxtimplc, the technique wiL\ \uccc\\t’ully used for the in[erpolatlon

• f ~ regular grid nf 31>

warmed dirto in “Star Trek IV: The Voyage Home” I I3). Methods tt)r oul(muitic:illy gener:itin.g ctmwptmdlng vertices orpol}gon~ for lnicrpol:ili(m ha~c been dcteloped. [.$1161 [)thcr cnmputcr gr:iphics techniques which ctin hc uwd for object mctarnnrphosii include WIid def(mnati[ms I I j [ l?] and purtlcle i} s[em~ {I()). In cuch u:iw ihc 31) model of the first object ii trwwf(mned I{)h:i~e the shape wrd Surt,icc prnpertie~ of the wcond

• mwicl. and the rcwlting

imimutilm is rcnderwl :md rec(mied. 2.3 2D Computer Graphics Techniques While three-ciirllcn~ion:tl ohjwt rnctumorphoiis i\ Anatural wdutmn whcn both (~hjccti tire cwil} rnodelcd for (hc cornputcr. ()!tcn the complexly

• f the wbjccts

makes this tipprntich imprxtical. F“nr e~imlplc. men} tipplications ot the CI”!CCI require lrwrif{mrn;i[i(ms hetwcen c(mlplcx ohject~ wch :ii anirnfili. In this caw it is often cwicr to m:inipulate wwrned phot~)griiphf of the went u~lrlg IWO dirncnsi(mul image pmccssing techniques than to attempt to model ;ind render the dct:lils of the anirn:il’s tippewmce for rhe computer. The stmplcit method for changing tmc digital image Into another i~ \Impl> to croswliswl~c Iwv.wn

• Ihcm. The colnr nf c:ich prxcl if

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