The web version of this article is available at the following link.
As a guest, you are not allowed to view links.
Register or
LoginMotion blur analysis shows that Z-318 is a composite of images photographed by cameras with differing motions while comparison of Z-317 with Z-318 disallows the images of Toni Foster as consecutive.
Z-317 shows three whitish spots on top of the partition between the operator and the passenger compartments of the limousine. A strong motion blur on Z-318 transformed these spots into three diagonal streaks.
A clockwise rotation of the camera about a center of rotation below and to the right of the diagonal streaks could have produced this motion blur of Z-318. Alternately a counterclockwise rotation of the camera about a center above and to left of the diagonal streaks would explain this motion blur. In both cases the center of rotation would be along a line perpendicular to the direction of the diagonal streaks. The strength or the relative stretching of the blurred object would be proportional to distance of the object from the center of rotation. The two preceding comments describe the characteristic features of a rotational motion blur.
A translational motion of the camera down and to the right during exposure of Z-318 would have transformed the three whitish spots of Z-317 into the three diagonal streaks. Under these conditions the translational motion of the camera would have elongated all objects to the same extent and directed opposite of the camera’s movement.
Z-317 shows the windshields of two motorcycles. One windshield appears just behind the officer’s helmet and the second windshield is in front of the officer. Each windshield has a pair of nearly horizontal highlights. The motion blurs of Z-318 diagonally transform these highlights similar to the transformation of the whitish spots on the partition into diagonal streaks.
The motion blur of Z-318 changes the shape of the officer’s helmet. However, neither the extent nor the direction of the motion blur is readily discernible. So the highlights of the windshields are the chosen details for measurement of the motion blurs.
The direction of the diagonal streaks near the lower-right of Z-318 nearly coincide with the direction of the motion blurred highlights at the lower-left side. This near coincidence of directions of motion blurs near opposite sides of the Z-318 is decisive evidence that the camera movement during exposure was predominantly a translation.
Z-317 shows the forearm of Toni Foster resting horizontally across her mid section. The shape of the brighter portion of the arm is roughly rectangular. The strong diagonal motion blur on Z-318 that affected the partition and the windshields should transform the rectangle into an irregular hexagon resembling a quadrilateral. However, Z-318 shows the arm with practically the same shape and size as on Z-317. This failure of the strong translational motion blur to affect all objects exposures Z-318 as a composite.
Frames Z-317 and Z-318 portend that the forearm of Toni Foster rotated approximately 30 degree in 1/18 second. Neither frame shows a rotational motion blur of the arm. So this pair reports that the forearm accelerated from rotational rest, reached a maximum angular speed then decelerated to rotational rest during the 1/36 second interval between the two exposures. These humanly impossible motions expose the images of Toni Foster on Z-317 and Z-318 as not chronologically ordered.
Appendix 1 - Computer Generated Motion Blur of Toni Foster
On the untouched Z-317, three objects are useful in evaluating the effects of a diagonal motion blur. These objects are the whitish square at the bottom of Foster’s right leg, the shape of her upper head and the roughly rectangular arm.
Paint Shop Pro 8.0 generated this translational motion blur of Toni Foster. This simulated image shows that the diagonal motion blur changed shapes of her features. In particular the motion blur diagonally elongated the whitish square at the bottom of the left foot. This change is reminiscent of the transformation of the whitish spots on the partition into elongated streaks. The change in shape of Foster’s head is similar to the change in shape of the officer’s helmet. However, the blurring of the simulated arm of Toni Foster is strikingly different from the shape of the arm on Z-318. Further Z-318 show two white spots representing each foot of Toni Foster whereas the simulated image shows only one foot. A careful comparison of the Z-317 with Z-318 reveals that the positions of Foster’s legs on Z-318 do not correspond with their positions on Z-317.
Appendix 2 - Analysis of Camera Motion at Two of Three Objects
Place a Cartesian coordinate system at the lower left corner of Z-318. Define F = a i +b j as the position vector of Toni Foster, P = c i + d j as the position vector of the partition and W = e i + f j as the position vector of the windshield. Let R = x i
+ y j be the position vector of center of camera rotation and let T = (T Cos t) i + (T Sin t) j be a constant translation vector of magnitude T at an angle t.
For a rotation of the camera through an angle r, the rotational component of the camera motion at Foster is Sf = Tan (r) [ (y-b) i + (a-x) j ]. Likewise the rotational component at the partition is Sp = Tan (r) [ (y-d) i + (c-x) j ] and the rotational component at the windshield is Sw = Tan (r) [ (y-f) i + (e-x) j ]. These equations recognize the straightness of the motion blurs on Z-318 and use tangents to approximate the arcs of the rotational components.
The composite motion of the camera at Foster, Cf is null while the composite motion at the partition vector Cp has a magnitude Cp at an angle p. Similarly the composite motion at the windshield, vector Cw has magnitude Cw at an angle w. The angles p and w are measured as counterclockwise rotations from the positive x axis.
Equating the vector sum of the rotational component at Foster and the common translational motion to the null composite motion gives equation one.
As a guest, you are not allowed to view links.
Register or
LoginLikewise taking the vector sum of the rotational motion at the partition and the common translational motion gives equation two for the composite motion at the partition.
As a guest, you are not allowed to view links.
Register or
LoginSimilarly equation three states that the vector sum of the rotational motion at the windshield plus the translational motion equals the composite motion at the windshield.
As a guest, you are not allowed to view links.
Register or
LoginSubtracting equation one from equation two and subtracting equation one from equation three eliminate T, t, x and y from this system.
As a guest, you are not allowed to view links.
Register or
LoginBoth sides of a vector equation have the same direction. So the magnitude of the j
component divided by the magnitude of the i component on the right side of the equation equals the corresponding ratio on the left side.
As a guest, you are not allowed to view links.
Register or
LoginThe relationships of equations five restrict the direction of the composite motion at the partition to a perpendicular to the displacement of the partition from Toni Foster. Correspondingly the direction of the composite motion at the windshield is perpendicular to the displacement of the windshield from Toni Foster.
Taking the vector dot product of each member of equations four with itself reveal constraints upon the rotation angle, the composite motions and the distances of Toni Foster from the other two objects.
As a guest, you are not allowed to view links.
Register or
LoginVerbally equations six state that the magnitude of the composite motion at the partition divided by the distance of the partition from Foster equals the magnitude of the composite motion at the windshield divided by the distance of windshield from Foster and equals the tangent of the rotation angle of the camera.
From equation one the direction of the translational motion of the camera is given by equation seven.
As a guest, you are not allowed to view links.
Register or
LoginSo the direction of the common translational motion of the camera is perpendicular to the displacement of Foster from the center of rotation. The magnitude of the translation is constrained by the relationships of equations eight.
As a guest, you are not allowed to view links.
Register or
LoginGeometrically equation eight states that the magnitude of the translational motion divided by the distance of Foster from the center of rotation equals the magnitude of the composite motion at the partition divided by the distance of the partition from Foster and equals the magnitude of the composite motion at the windshield divided by the distance of the windshield from Foster.
