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CSE486, Penn StateRobert CollinsLecture 22:Camera MotionReadings: T&V Sec 8.1 and 8.2CSE486, Penn StateRobert CollinsMoving CamerayawpitchrollTxTyTzFrom one time to the next, the camera undergoesrotation (roll, pitch, yaw) and translation (tx,ty,tz)tt+1t+2t-1t-2t-3t-4Camera takes a sequence of images (frames)indexed by time tCSE486, Penn StateRobert CollinsMotion (Displacement) FieldsTime tTime t+1Displacement fieldCSE486, Penn StateRobert CollinsMotion Field vs Optic FlowMotion Field: projection of 3D relative velocity vectors onto the 2D image plane.Optic Flow: observed 2D displacements of brightness patterns in the image.Motion field is what we want to know.Optic flow is what we can estimate.CSE486, Penn StateRobert CollinsMotion Field vs Optic FlowSometimes optic flow is a good approximationto the unknown motion flow.optic flow fieldWe can then infer relative motion between the camera and objects in the world.CSE486, Penn StateRobert CollinsWarning: Optic Flow ≠ Motion FieldConsider a moving light source:MF = 0 since the points on the scene are not movingOF ≠ 0 since there is a moving pattern in the imagesCSE486, Penn StateRobert CollinsMotion FieldWe are going to derive an equation relating 3D scene structure and velocity to the 2D motion flow field.CSE486, Penn StateRobert CollinsMotion FieldWhat is a Field anyways?Image a vector at each point in space. This is avector field. In 3D space, we will look at the fieldof 3D velocity vectors induced by camera motion.In 2D, we will be looking at the projections of those3D vectors in the image. There will be a 2D flow vector at each point in the image. This is the 2Dmotion flow field.CSE486, Penn StateRobert CollinsRecall : General Projection EquationRelative R,TProjectionInternalparamsStrategy:1) Assume internal params known (set to identity)2) At time t, set R=I, T=03) At time t+1, movement generates a relative R and T simplifying assumption: small motion --> small rotation4) Compute 3D velocity vector 5) Compute 2D velocity vector a function of X,Y,Z,R,T,fCSE486, Penn StateRobert CollinsDisplacement of 3D World PointTime t: 3D position PTime t+1: 3D position RP+T3D Displacement = RP+T - PNow consider short time period (like time betweentwo video frames = 1/30 sec). Can assume a smallrotation angle in that amount of time. Make a smallangle approximation and rewrite displacement. In thelimit (infinitesimal time period), we will get a velocity.CSE486, Penn StateRobert CollinsWrite Rotation matrix in terms of Euler AnglesCSE486, Penn StateRobert CollinsCSE486, Penn StateRobert CollinsSmall Angle Approxfor small angle x xx ≈sin1cos ≈xx0sinsin ≈yxCSE486, Penn StateRobert CollinsCSE486, Penn StateRobert Collins3D VelocityUnder small angle approx,displacement = RP+T - P = (I+S)P+T-P = SP+TNote: SP = [θx, θy, θz]T X PIn limit, displacement becomes a velocityV = T + ω X P3D velocitylinear velocityangular velocitywhereCaution: Abuse of notation ==> we are re-using T as a velocityCSE486, Penn StateRobert Collins3D Relative VelocityffPPVVppvvZZzzThe relative velocity of P The relative velocity of P wrt wrt camera:camera:Camps, PSUNote: we change signs of the velocity to beconsistent with the book. This should notcause concern. It just depends on whetheryou want to think of the motion as being dueto the camera or the scene.CSE486, Penn StateRobert Collins3D Relative VelocityffPPVVppvvZZzzThe relative velocity of P The relative velocity of P wrt wrt camera:camera:Camps, PSUCSE486, Penn StateRobert Collins3D Relative VelocityffPPVVppvvZZzzWhere are we in this lecture?• We just derived an equationrelating R,T and to the 3D velocity vector at each scenepoint.• We can think of that velocity as a little vector inthe scene.• Now ask, what does the projection of that vectorlook like in the image? It is a 2D vector. It is oneof the vectors that make up the Motion Field!CSE486, Penn StateRobert CollinsMotion Field: the 2D velocity of pffPPVVppvvZZzzTaking derivative Taking derivative wrt wrt time:time:Camps, PSUPerspectiveprojectionCSE486, Penn StateRobert CollinsMotion Field: the velocity of pffPPVVppvvZZzzCamps, PSUCSE486, Penn StateRobert CollinsMotion Field: the velocity of pffPPVVppvvZZzzCamps, PSUCSE486, Penn StateRobert CollinsMotion Field: the velocity of pffPPVVppvvZZzzCamps, PSUTHIS IS THE EQUATION WE WANT!!!!CSE486, Penn StateRobert CollinsMotion Field: the velocity of pffPPVVppvvZZzzTranslationalTranslationalcomponentcomponentCamps, PSUCSE486, Penn StateRobert CollinsMotion Field: the velocity of pffPPVVppvvZZzzRotationalRotationalcomponentcomponentNOTE: The rotational component is independent of depth Z !NOTE: The rotational component is independent of depth Z !Camps, PSUCSE486, Penn StateRobert CollinsSpecial Case I: Pure TranslationAssume Assume TTzz ≠≠ 0 0Define:Define:Camps, PSUCSE486, Penn StateRobert CollinsSpecial Case I: Pure TranslationWhat if What if TTzz == 0 ? 0 ?All motion field vectors are parallel to each other andAll motion field vectors are parallel to each other andinversely proportional to depth !inversely proportional to depth !Camps, PSUTIE IN WITH SIMPLE STEREO!CSE486, Penn StateRobert CollinsSpecial Case I: Pure TranslationThe motion field in this case is RADIAL:The motion field in this case is RADIAL:••It consists of vectors passing through pIt consists of vectors passing through poo = (x = (xoo,,yyoo))••If:If:•• TTzz > 0, > 0, (camera moving towards object)(camera moving towards object)•• the vectors point away from p the vectors point away from poo••ppoo is the is the POINT OF EXPANSIONPOINT OF EXPANSION•• TTzz < 0, < 0, (camera moving away from object)(camera moving away from object)••the vectors point towards pthe vectors point towards poo••ppoo is the is the POINT OF CONTRACTIONPOINT OF CONTRACTIONTTzz > 0> 0TTzz < 0< 0Camps, PSUCSE486, Penn StateRobert CollinsPure Translation:Properties of the MF• If Tz≠ 0 the MF is RADIAL with all vectorspointing towards (or away from) a singlepoint po. If Tz = 0 the MF is PARALLEL.• The length of the MF vectors is inverselyproportional to depth Z. If Tz ≠ 0 it is alsodirectly proportional to the distancebetween p and po.Camps, PSUCSE486, Penn StateRobert CollinsPure Translation:Properties of the MF• po is the vanishing point of the direction oftranslation.• po is the intersection of the ray parallel tothe


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PSU CSE/EE 486 - Camera Motion

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