algorithm - How to mathematically obtain camera trace path from video -

i have aerial video of (approximately flat) view (no depth maps required).

i want obtain shape of ground path camera center passed upon.

the path not have correct correspondence worl'd north, or have world coordinates. have approximately same shape, if affine transformed.

i have each frame, matrix represents affine transformation between frame , previous frame, @ accuracy, meaning have matrix each pixel in frame, gives location of pixel in consecutive frame.

given camera moving, how can obtain shape (over time) of ground path of spot on ground in center of frames (=the trace)?

more generally, question knowing relative center of frame, each frame, correspondence first frame.

any leads, answers benefit me great deal.

let's call transformation between frame i , i+1 t. let's @ point p[i] = (xi,yi) in frame i. if want know point in frame i+1 need apply transformation t on point p[i] , new point p[i+1] = t(p).

in order path camera did start frame 1 center point of image p1 = (row/2,col/2) , calculate p2 = t(p1). path camera did between frame 1 , 2 approximately line connects between center of frame 2 , p2. can keep doing frame 2 calculating p3 = t(p2) , on... problem points in in image coordinates , not world coordinate.

to solve decide our "world coordinate system" pw1 = (0,0) center of first frame. center of camera in second frame in our new "world coordinate system" pw2 = ((row/2,col/2) - p2) - pw1. third frame center pw3 = ((row/2,col/2) - p3) - pw2 , i-th frame pwi+1 = ((row/2,col/2) - pi+1) - pwi. idea pw_new = movement_of_the_camera_between_frames - last_location.

in end have set of points pw1,pw2,..,pwn path camera center did. fit spline or , shape of path.