This option allows for crossing light rays in a relatively simple manner. The correction works reasonably well with light from a large apperture source (such as a linear halogen photographic lamp), but is not as successful with light from a small source (such as a laser or arc lamp) where the light rays do not cross (in which case [D Diverging, attenuating light rays] is more appropriate).
In principle the intensity Pr(x,y,é) of a light ray at angle é to the x axis decays as
dPr = -A(C) Pr ds, (1)
where A(C) is the attenuation rate as a function of concentration C=C(x,y) for which the dye is a nominal marker (and hence A=A(C)=A(x,y)), and ds=dy/siné is the incrimental distance a ray of light travels. The overall intensity of light at a particular point, defined by
ôã
P(x,y) = ³ Pr(x,y,é) dé, (2)
õ0
behaves as
dP ôã dPr
ÄÄ = ³ ÄÄÄ dé
dy õ0 dy
ôã dPr
= ³ ÄÄÄ cosecé dé
õ0 ds
ôã
= - ³ A(C) Pr cosecé dé. (3)
õ0
For weak concentrations we may take
(4)
A(C) = a C,
where a is a constant (the concentration C is independent of angle) so that
dP ôã
ÄÄ = - a C ³ Pr cosecé dé. (5)
dy õ0
The finite difference approximation with incriments ëI and ëy allows this equation to be rewritten as
ôã
ëP = - ëy a C ³ Pr cosecé dé, (6)
õ0
in which we may replace cosecé dé with an integral in x to give
ôx+ì dx'
ëP = ëy a C ³ Pr(x',y-ëy,é) ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ. (7)
õx-ì û[(x-x')ý + ëyý]
Without loss of generality we may write
Pr(x,y,é) = P(x,y) T(x,y,coté), (8)
where T(x,y,coté) gives the relative amplitude of the different rays passing through a point x,y at an angle é and from the definition of P(x,y)
ôx+ì
³ T(x,y,x'/ëy) dx' = 1. (9)
õx-ì
This may be substituted into equation (7) to yield
ôx+ì dx'
ëP = ëy a C ³ P(x',y-ëy) T(x',y-ëy,(x'-x)/ëy) ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ.
õx-ì û[(x-x')ý + ëyý]
(10)
To simplify the image correction problem we shall assume that T/û[(x-x')ý+ëyý] can be split into filter and amplitude components. The filter component F(x,y,x') depends only on the geometric properties of the light sheets and may be approximated in a simple manner by a low pass filter combined with a translation allowing for the nominal ray direction. This filter component caters for the divergence and crossing of the light rays. For brevity we shall write this filter operation as
ôx+ì
f(P,x,y) = ³ P(x',y-ëy) F(x,y,x') dx'. (11)
õx-ì
The amplitude component G(x,y) parameterises the decay of the illumination due to divergence and absorbtion of the medium (excluding the fluorescent dye). This amplitude component will be obtained from a calibration image of uniform dye concentration. Using the above assumption equation (10)
ëP(x,y) = ëy a C(x,y) G(x,y) f(P,x,y). (12)
Suppose P0, C0 are the intensity and concentration fields for a calibration image of uniform concentration. From this calibration
G(x,y) = ëP0(x,y) / [ëy a C0 f(P0,x,y)], (13)
so that (12) may be written as
C(x,y) f(P,x,y)
ëP(x,y) = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ëP0. (14)
C0 f(P0,x,y)
Following the linear attenuation approximation, we assume the observed intensity field p(x,y) is related to the concentration and illumination by
p(x,y) = à C(x,y) P(x,y). (15)
Substituting this into (14) we may see that
f(p,x,y)
ëP(x,y) = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ëp0, (16)
à C0 f(p0,x,y)
which provides the method used to determine the intensity of the light sheet in this option. Finally, once P(x,y) is determined by stepping (16) through an image, we may determine the concentration from
C(x,y) = p(x,y)/[à P(x,y)] (17)
A knowledge of the fluorescing efficiency à is not required unless the actual intensities within the sheet are required. The actual concentration of the fluorescent dye constituting a nominal unit concentration (as defined implicitly by C0) may be chosen to optimise the quality of the images.
NOTE: Before applying this correction, the image intensities should be mapped onto a linear intensity scale passing through zero.
The Cursor submenu is produced to allow each entry point for a diverging
light ray (such as that marked by the shadow cast by some slide or mask)
to be specified. The point chosen should be at or near the boundary of the
image through which the light ray enters. For assistance on the use of the
cursor, consult [H Help] within the cursor submenu.
This selection allows the orientation and spread of the divergence
triangle to be changed. The left and right or up and down arrow keys move
the triangle in the appropriate direction. <C> reduces the width of the
triangle while <E> enlarges the width. <S> may be used to
toggle between steps of one and sixteen pixels at a time. Once the
triangle is positioned, <Q> continues execution. From a command file it is
more convenient to enter the rays manually using <M> which prompts for the
angle to the nominal ray direction and the spread angle.
{If ray direction entered manually}
{If ray direction entered manually}
Between four and ten ray triangles may be specified. They will
subsequently be used to generate a least squares mapping of the spread of
the light rays. The first three times this prompt is produced only the
"yes" answer <Y> will be accepted, thereafter either yes <Y> or no <N> is
valid.