Because raw images reflect the nonuniformity and nonlinearity of the detector response, a photometric correction is performed on a pixel-by-pixel basis so that pixel DN values are replaced by normalized fluxes (flux numbers, or FN) which are linearly related to the incident photon flux on the vidicon faceplate. For high dispersion images, each pixel value within a circle (as mapped into geometrically correct space) encompassing the useful portion of the image is converted to FN; pixels outside the circle are left with uncorrected (DN) values. In low dispersion, the corrected portion of the image is further restricted to a swath encompassing the spectral order. Detailed data defining these regions and the photometric correction process itself are contained in the following subsections.

The intensity transfer function (ITF) contains the information needed to map DN to FN on a pixel-by-pixel basis across an IUE image. Given a pixel with a particular DN value, the ITF is used, in the manner described in Section 5.3, to obtain the corresponding linearized FN value.

The ITF for each camera is defined in geometrically correct space and is generated from a series of geometrically corrected mercury flood-lamp flat-field images at graded exposure levels (11 levels for SWP; 12 levels for LWR and LWP). for the SWP and LWR cameras, each exposure level in the ITF is constructed of an average of several individual images from which reseau marks have been removed by the REMRES program described in Perry and Turnrose (1977). The lowest level is the zero-exposure level ("null", or "pedestal") representing the background left by the vidicon camera flood/erase prep procedure which precedes every normal exposure. The highest level is a several-hundred-second flood-lamp exposure which reaches the vidicon saturation limit of 255 DN over part of the image; less sensitive areas remain unsaturated, although over most of the tube in SWP and LWR, a DN of 220 is reached or exceeded. Holm (1979) and Turnrose (1980) have discussed this aspect of the ITF in the context of low dispersion spectra, where the least sensitive areas of the tube correspond to the longer wavelengths in LWR and the shorter wavelengths in SWP. Table 5-1a, reprinted from Holm (1979), lists the appropriate upper DN limits of the current SWP and LWR ITFs in the vicinity of the low dispersion spectral orders. Table 5-1b, reprinted from Imhoff (1984b), lists the maximum DN levels for an LWP low dispersion spectrum in which the best photometric accuracy of the ITF is preserved. As described in Section 5.3, input image DN levels which exceed the top level of the ITF are now converted to FN by extrapolation.

Maximum DN Levels for Best Accuracy

In LWP Low Dispersion Spectra

LWP | |
---|---|

DN | |

2100 | 205 |

2300 | 220 |

2500 | 240 |

2700 | 245 |

2900 | 245 |

3100 | 245 |

3300 | 245 |

Intermediate ITF levels are generated from intermediate-length exposures in order to map the camera response through the mid-range of DN values and are spaced sufficiently closely to allow the conversion to FN by linear interpolation between the levels as described in Section 5.3.

An exposure value (FN value) is associated with each level of the ITF. This
exposure value is normalized in an arbitrary (but fixed) way so that the FN
associated with the highest level of the ITF is near 20,000 (for SWP the
highest ITF flux level is 17632; for LWR, 25220; and for LWP, 19784). The FN
associated with level i of the ITF for camera k is

T(k,i) MULT(k,i)

FN(k,i) = -----------------

FACTOR(k)

FN(k,i) = -----------------

FACTOR(k)

where T(k,i) is the effective exposure time (in seconds) for level i and camera k, MULT(k,i) is a scaling factor for level i and camera k, and FACTOR(k) is a level-independent scale factor for camera k. Although in principle MULT(k,i) can vary from level to level, (e.g., to account for flood- lamp variability), in practice MULT(k,i) is used as a function of k only, and lamp drifts during the creation of the ITF are accounted for in adjustments to the effective exposure times T(k,i). Tables 5-2 and 5-3 list the effective exposure times and the scale factors for the three ITFs in use as of November 1981. (These ITFs are designated LWP ITF1, LWR ITF1, and SWP ITF1). With these data and equation 5-1, the exposure value of each ITF level is determined.

LWP | LWR | SWP | |
---|---|---|---|

MULT | 17.00 | 17.00 | 11.00 |

FACTOR | 0.28333 | 0.28333 | 0.1778 |

As is pointed out in Section
9, the T
(in units of 0.01 seconds) and MULT
values pertinent to the ITF actually applied to a given image are documented
in the processing history portion of the image header labels of the
photometrically corrected image and all derived files.

Although as mentioned in Section 5.2.2 the highest exposure levels in the ITFs reach the saturation limit of 255 DN over portions of each camera, ITF DN values about 250 present potential hazards due to "hidden" saturation effects. Whitman, Bohlin, and Turnrose (1981) discuss the fact that because of noise in the individual flat-field images of which the ITF is composed and the fact that the production ITFs are composed of averages of several independent (and geometrically corrected) images, even final ITF values several DN below 255 may have been influenced by saturation in some of the raw constituent images.

To alleviate such potential hidden saturation effects, the DN value of 250 has been defined as the maximum valid ITF DN level. This in essence means that DN levels >250 in the ITF are ignored (except in the pathological case (discussed in Section 5.3.2.2.2) in which there are insufficient valid ITF levels for the pixel in question). More specifically, this means that DN > 250 in spectral data images will be converted to FN by extrapolation from lower values, and that ITF extrapolations will be based only on points with DN <= 250; see Section 5.3.2.2.

Pixels in the raw image are converted from DN to FN by the program PHOTOM only within the portion of the image relevant to the spectral data. In the case of high dispersion, this means that only pixels within a circle slightly smaller than the target boundary are converted; pixels outside the circle are left unconverted. Note that this circle is defined in geometrically correct space, hence the photometrically corrected region will not be bounded by a true circle when viewed in the raw geometry. Table 5-4 lists the center and radius values of these circles for the three operational cameras.

Center | |||
---|---|---|---|

Camera | Line | Sample | Radius |

LWP | 400 | 390 | 347 |

LWR | 395 | 402 | 350 |

SWP | 390 | 390 | 358 |

*In geometrically correct coordinates |

In low dispersion, it is unnecessary to convert the entire region within the circles defined above, so only a swath or band encompassing the spectral orders is converted. This band, oriented parallel to and centered between the large and small-aperture spectra, has a width in the sample direction of 320 pixels. Note that in previous documentation this width has been incorrectly specified as 160 pixels. Although the 160-pixel width would be sufficient to encompass the line-by-line data extracted in low dispersion (see Section 7), the larger area has been retained to allow the conversion of sufficient portions of the tube to facilitate potential studies of the tube background.

In August 1982, a further restriction to the area photometrically corrected in the photometric correction in low dispersion is done only within the camera-dependent rectangular area defined by the standard partial read parameters; pixels outside this area are left in DN units. Table 3-1 presented in Section 3.3 lists these parameters, which apply directly to raw-image space. Note that this limitation on the photometric correction is applied to all low dispersion images, whether or not they are partial reads. The only effect on full-frame low dispersion images, however, is a slight truncation of certain corners of the photometrically corrected swath; no extracted spectral data are affected.

Since the ITFs are defined in geometrically correct space, the conversion of a
pixel DN value in a raw image to FN requires first that the corresponding
position in the ITF be determined; conceptually, this is done via the
geometric mapping function G discussed in Section
4.4. (In practice, for
computational simplicity the mapping is actually done in the reverse direction
(geom-to-raw), using the G^{-1} function).
Since in general the position in
geometrically correct space resulting from this mapping will not coincide with
an integral pixel in the ITF, the ITF must be spatially interpolated. To do
this, the input pixel DN value is first used to calculate four FN values using
the ITF curves of the four pixels surrounding the mapped location; details of
such FN assignments are addressed in Section
5.3.2.
A flux value representing
the final FN corresponding to the input pixel is then computed from the
bilinear spatial interpolation of these four neighboring FN values.

Following Lindler (1982a), this process is described mathematically below.
Given a pixel with value DN at the location in the raw image (s,l), its
position (x,y) in the geometrically corrected ITF is

(x,y) = G(s,l)

where G is defined in Section
4.4.
If the function IFIX(x) returns the
largest integer <= x, then define
If we now represent the FN-assignment function (described in Section 5.3.2) for a pixel at location (X,Y) with value DN by F(X,Y,DN), then the four neighboring flux values to be used in the spatial interpolation are

F1 = F(X,Y,DN)

F2 = F(X,Y+1,DN)

F3 = F(X+1,Y,DN)

F4 = F(X+1,Y+1,DN).

Finally, the interpolated FN value is calculated as below.
If
F2 = F(X,Y+1,DN)

F3 = F(X+1,Y,DN)

F4 = F(X+1,Y+1,DN).

u = x - X

v = y - Y

then

FN = (1-u)(1-v)F_{1} + (1-u)vF_{2} + u(1-v)F_{3} + uvF
_{4}.

In this subsection the techniques used to evaluate the function F(X,Y,DN) defined in Section 5.3.1 are described. F(X,Y,DN) represents the conversion of DN to FN for the specific integer-pixel location (X,Y) in the geometrically correct frame of reference. The evaluation of F(X,Y,DN) proceeds in one of several ways using either interpolation or extrapolation depending on where the DN value falls with respect to the ITF for that pixel.

In most instances when the input DN is less than or equal to the maximum valid ITF DN value of 250 (see Section 5.2.2.2), an FN value is assigned by simple linear interpolation between the flux values of the various levels of the ITF. That is, the input DN is compared linearly to the DN corresponding to each level of the ITF, and the associated flux is interpolated accordingly. The majority of the pixels in a well-exposed image will fall below DN = 250 and will thus be converted to FN by this simple interpolation process.

There are three instances, however, in which the input DN may be less than or equal to 250 but extrapolation rather than interpolation will be used. These are when:

- the input DN exceeds the highest DN in the ITF for that particular pixel (Section 5.3.2.2.2), or
- the input DN falls between two ITF levels, the higher of which exceeds the maximum valid DN limit of 250 (Section 5.3.2.2.2), or
- the input DN falls below the lowest DN in the ITF (null level) for that particular pixel, in which case extrapolation to negative FN value is made (Section 5.3.2.2.1).

Extrapolation to negative FN values occurs in all instances in which the input
DN falls below the null-level of the ITF. This can result from statistical
fluctuations or null drift in which the null level of the particular input
image being photometrically corrected falls below the ITF null level as a
whole. In this situation FN values are assigned by simple __linear
extrapolation__ from the first two points of the ITF (null level and lowest
finite-exposure-time level). The current software can accommodate
extrapolation down as far as FN = -3488.

Extrapolation to positive FN values occurs in all instances in which:

- the input DN exceeds the overall maximum valid ITF DN level of 250 (the special case of saturation is discussed further in 5.3.2.2.3), or
- the input DN exceeds the highest DN level in the ITF for that pixel, or
- the input DN falls between two successive ITF levels, one of which
is valid (DN <= 250) and one of which is invalid (DN > 250).
In each of the above instances, an FN is assigned on the basis of a

__linear least squares fit__to the highest three valid ITF levels, as described in Whitman, Bohlin, and Turnrose (1981). The only exception is when there is not a sufficient number (three) of valid ITF levels on which to base an extrapolation. In such a case, the FN value is either assigned the flux number of the lowest level, if only one valid point is found, or, in the case of two valid points, the FN is assigned by linear extrapolation. Such situations, although unlikely, could conceivably occur for a "hot" pixel for which ITF levels above the second or third level are invalid. Extrapolation of unsaturated pixels may reach up to FN = 65536, the maximum flux value which can be accommodated under the coding system used for pixel values in photometrically corrected images (see Section 5.3.3). Extrapolations which would otherwise exceed this limit are referred to as "excessive extrapolations" and are cut off at FN = 65534 and flagged by the coding system in the same way as saturated pixels. The limit of 65534 for excessive extrapolations instead of the seemingly more logical value of 65536 is necessitated by the FN coding system.### 5.3.2.2.3 Saturated Pixels

Extrapolation of FN for saturated pixels represents a special case of the extrapolation described in Section 5.3.2.2.2, in the sense that the extrapolated FN is limited to a maximum value of 65534 and is flagged in such a way (Section 5.3.3) as to distinguish the FN values from those determined for unsaturated pixels.

### 5.3.3 CODING OF FLUX VALUES IN PHOTOMETRICALLY CORRECTED IMAGE FILE

The current software allows a considerably more extensive flagging system for exceptional pixels than did the old reduction software. In order to accommodate this additional flagging, however, a complex relationship between the true pixel FN value and the representation of this value in the photometrically corrected halfword image ("PI") file is necessitated. This relationship or coding system is such that the range of coded values in the image, Fimage, is itself used to flag the special cases of extrapolation, saturation, or no photometric correction, as shown in Table 5-5 , taken from Turnrose, Bohlin, and Lindler (1980). The coded Fimage values are placed in the PI file in a halfword format (16 bits per pixel, values in the range 32767, with negatives expressed in two's complement form). As used in the IUESIPS processing, the actual FN values are recovered, at the time of spectral extraction, from the coded Fimage values by the inverse relationships listed in the last column of Table 5-5.

## 5.4 LIMITATIONS TO PHOTOMETRIC LINEARITY

The techniques of photometric correction described in the preceding sections utilize ITFs constructed from flood-lamp exposures as discussed in Section 5.2.2. The most sensitive tests of the overall photometric linearity of the IUE system as it relates to user data are those which analyze the linearity of spectra extracted from well-controlled graded exposures of standard stars. Holm (1982) has briefly summarized the linearity errors measured in such a way. He notes the most significant errors as follows. SWP has linearity errors in the range -10 to -20 percent for net DN <= 20, and +10 to +15 percent for net DN > 220 at 1300 A; LWR has linearity errors of +10 to +20 for net DN <= 40; and LWP has linearity errors of -5 to -10 percent for net DN < 50.

In the case of the SWP, Holm (1981) found no improvement in the photometric behavior at high DN levels when a test version of the ITF incorporating a twelfth level was used, implying that at least in this case, the extrapolation from an eleven-level ITF was no worse than interpolation within a twelve-level ITF. For the LWR, the linearity errors are not constant across the camera, so that simple adjustments of the ITF effective exposure times, for example, would not reduce the errors in all cases.

Recent linearity studies are summarized in Oliversen (1984a,b). If new ITF calibrations are performed on the cameras, their linearity properties will differ from those described above.