U/W Photographs and Video Bitmaps Image Area Calculation

Figure 9. Towcam image area schematic

The height or Y – axis of the image corresponds to the along track direction of TOWCAM and the width or X-axis corresponds to the cross-track distance. The following variables were used to derive the width (W) and height (H) of a pixel at the midpoint of the image and then the image area:

A         = height of camera above the sea floor in metres

AR       = vertical to horizontal aspect ratio of a pixel in the digitized image

β          = viewing angle in the vertical dimension of the image

Θ         = vehicle pitch angle + camera axis tilt

φ          = vehicle roll angle

D_L        = actual distance between laser points in metres

X_L        = projected distance between laser dots on sea floor in metres

XP_L      = distance between laser dots in pixels

XP       = width of total image in pixels

YP       = height of total image in pixels

YP_L      = distance from top of image to laser dots in pixels

W        = width of a single pixel at the midpoint of the image in metres

H         = height of a single pixel at the midpoint of the image in metres

Many image analysis softwares (e.g. Image Pro Plus) can be used to determine the pixel dimensions of the image as listed above. TOWCAM contains roll and pitch sensors and an altimeter (Gordon et al 2006) which determine Θ, φ and A respectively. For the still camera, photographing a 0.5m ring at various distances in water derived the angle β. Bradford (2005) derived it for video cameras by imaging a calibration grid in air then adjusting the result for the index of refraction difference in water.

Camera/sea floor geometry in a plane orthogonal to the sea floor along the Y – axis of the image, for the case where the laser dots are located above the midpoint of the image (YP­­­_{L ­­} < YP/2). The horizontal axis is in pixels, not actual distances on the image.

If the sea floor is horizontal and flat over the image area then, with reference to the adjacent figure:

Φ = |((YP/2) - YP_L)| * β
YP

A₁ = A/cos(Φ +  Θ)

Note:   If the laser dots are below the midpoint of the image (YP_L>YP/2) then

A₁ = A/cos(Φ  -  Θ)

In either case

A2 = A/cos(Θ)

The distance between the laser dots on the sea floor (X_L) is a function of vehicle roll:

X_L = D_L / cos(φ)

If the line between the laser dots subtends an angle Ω at the camera in the X direction then

Ω/2 = arctan(X_L/(2*A₁))

and the equivalent distance between the laser dots when projected onto the midpoint of the image is

X_LM = 2 * A₂ * tan(Ω/2)

So the width of a single pixel at the midpoint of the image is:

W  = X_LM / XP_L

Now, if the camera is oriented vertically and the vertical to horizontal aspect ratio of a pixel is AR then the height dimension (H) of the pixel would be (W * AR). However, the camera is actually tilted forward by an angle Θ, thus

H = W * AR / cos(Θ)

Finally, if the image is N_X by N_Y pixels then the area of the sea floor encompassed by the image is:

Area = (N_X * W) * (N_Y * H)