Camera calibration & radiometryPhoto to learn namesToday’s classTranslationRotationFind the rotation matrixRotation matrixTranslation and rotationHomogenous coordinatesHomogenous/non-homogenous transformations for a 3-d pointHomogenous/non-homogenous transformations for a 2-d pointThe camera matrix, in homogenous coordinatesThe projection matrix for orthographic projection, homogenous coordinatesCamera calibrationIntrinsic parametersIntrinsic parametersIntrinsic parametersIntrinsic parametersIntrinsic parametersIntrinsic parametersExtrinsic parameters: translation and rotation of camera frameCombining extrinsic and intrinsic calibration parametersOther ways to write the same equationCalibration targetCamera calibrationCamera calibrationCamera calibrationCamera calibrationToday’s classIrradiance, ERadiance, LSolid angleWhat’s the solid angle subtended by this patch, area A, seen from P?Image irradiance/scene radiance relationshipHow the brightness depends on the surface properties: BRDFCoordinate systemHelmholtz reciprocity conditionHow does the world give us the brightness we observe at a point?Accounting for extended light sourcesSpecial case BRDF: Lambertian reflectanceShow surfacesCamera calibration & radiometry• Reading: – Chapter 2, and section 5.4, Forsyth & Ponce– Chapter 10, Horn• Optional reading:– Chapter 4, Forsyth & PonceSept. 12, 2002MIT 6.801/6.866Profs. Freeman and DarrellReq: FP 2, 5.4, H 10Opt: FP 4Req: FP 6Photo to learn namesToday’s class• First part: how positions in the image relate to 3-d positions in the world.• Second part: how the intensities in the image relate surface and lighting properties in the world.Translation=ZYxBBBBP=ZYxAAAAPAiˆAkˆAjˆBiˆBkˆBjˆPxAYAZAABOrHow does relate to ?PAPBABABOPP +=Rotation=ZYxBBBBP=ZYxAAAAPAiˆAkˆAjˆPxAYAZAHow does relate to ?PAPBPRPABAB =Find the rotation matrixProject onto the B frame’s coordinate axes.()=ZYXAAAAAAkjiOPˆˆˆAiˆAkˆAjˆPxAYAZA•••••••••=ZABYABXABZABYABXABZABYABXABZYXAkkAjkAikAkjAjjAijAkiAjiAiiBBBˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ•••••••••=ZABYABXABZABYABXABZABYABXABZYXAkkAjkAikAkjAjjAijAkiAjiAiiBBBˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ•••••••••=ZABYABXABZABYABXABZABYABXABZYXAkkAjkAikAkjAjjAijAkiAjiAiiBBBˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆRotation matrix•••••••••=ZABYABXABZABYABXABZABYABXABZYXAkkAjkAikAkjAjjAijAkiAjiAiiBBBˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆthisPRPABAB =implies•••••••••=ABABABABABABABABABBAkkjkikkjjjijkijiiiRˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆwhereTranslation and rotationLet’s writeas a single matrix equation:ABABABOPRP += −−−−−−−−=11000||1ZYXABBAZYXAAAORBBBHomogenous coordinates• Add an extra coordinate and use an equivalence relation•for 3D– equivalence relationk*(X,Y,Z,T) is the same as (X,Y,Z,T)• Motivation– Possible to write the action of a perspective camera as a matrixHomogenous/non-homogenous transformations for a 3-d point• From non-homogenous to homogenous coordinates: add 1 as the 4thcoordinate, ie• From homogenous to non-homogenous coordinates: divide 1st3 coordinates by the 4th, ie→1zyxzyxxyx1→zyTTzHomogenous/non-homogenous transformations for a 2-d point• From non-homogenous to homogenous coordinates: add 1 as the 3rdcoordinate, ie• From homogenous to non-homogenous coordinates: divide 1st2 coordinates by the 3rd, ie→1yxyx→yxzzyx1The camera matrix, in homogenous coordinates=TZYXfYXfZ010000100001• Turn previous expression into HC’s– HC’s for 3D point are (X,Y,Z,T)– HC’s for point in image are (U,V,W)→YXYXZffZHC Non-HCThe projection matrix for orthographic projection, homogenous coordinatesUVW          =100001000001          XYZT              →=YXTTYX1HC Non-HCCamera calibration• Use the camera to tell you things about the world. – Relationship between coordinates in the world and coordinates in the image: geometric camera calibration.– (Later we’ll discuss relationship between intensities in the world and intensities in the image: photometric camera calibration.)Intrinsic parametersForsyth&Poncezyfvzxfu==Perspective projectionIntrinsic parameterszyvzxu αα==But “pixels” are in some arbitrary spatial unitsIntrinsic parameterszyvzxu βα==Maybe pixels are not squareIntrinsic parameters00 vzyvuzxu+=+=βαWe don’t know the origin of our camera pixel coordinatesIntrinsic parameters00 )sin()cot( vzyvuzyzxu+=+−=θβθααMay be skew between camera pixel axes()PKzprrr01=Intrinsic parameters00 )sin()cot( vzyvuzyzxu+=+−=θβθαα−=1000100)sin(0)cot(1100zyxvuzvuθβθααUsing homogenous coordinates,we can write this as:or:()PKzprrr 0 1 =Extrinsic parameters: translation and rotation of camera frameWCWCWCOPRP += Non-homogeneous coordinates−−−−−−−−=11000||1ZYXWCCWZYXWWWORCCCHomogeneous coordinatesBlock matrix formCombining extrinsic and intrinsic calibration parametersForsyth&PonceOther ways to write the same