INNAME Input image (name) INCLASS (class) INSEQ 0.0 9999.0 (seq. #) INDISK 0.0 9.0 (disk unit #) IN2NAME Second input image (name) IN2CLASS (class) IN2SEQ 0.0 9999.0 (seq. #) IN2DISK 0.0 9.0 (disk unit #) OUTNAME Output image (name) OUTCLASS (class) OUTSEQ -1.0 9999.0 (seq #) OUTDISK 0.0 9.0 (disk unit #) BLC 0.0 4096.0 Bottom left corner of image 0=>entire image TRC 0.0 4096.0 Top right corner of image 0=>entire image IMSIZE 0.0 4096.0 Output image size in pixels APARM (1)=terp.ord., (2)=zero flag (3,4) = new reference pixel = (0,0) => center in output

HGEOM Type: Task Use: HGEOM will compute a non-linear geometric transformation of an input image using Everett interpolation (bilinear, bicubic, or biquintic). The transformation is implicitly defined by the headers of the two pictures. In short, HGEOM "makes this picture look like that picture". In other words, it transforms the first picture so that its geometry as described by the header is consistent with that of the second "template" picture. HGEOM is therefore useful for data comparison and will probably grow into a mosaicing task. HGEOM uses the 'scrolling buffer' concept of how to do geometric transformations. Blanks will appear in the output wherever the corresponding input pixel is outside the first image or blanked and the edges and blanked pixels expand by the width of the interpolation fucntion. Blanks will also appear if the internal buffer is exhausted but dynamic memory should ameliorate this. Warning messages are printed when pixels are blanked. ===================================================================== We recommend use of OHGEO instead of HGEOM. The latter propogates blanks and edges onto good areas by the width of the interpolation function, while OHGEO can even fill in small blanked regions and extrapolate a little. ===================================================================== Adverbs: INNAME.......Input map name. Standard defaults. INCLASS......Input map class. Standard defaults. INSEQ........Input map seq no. 0 => highest. INDISK.......Input map disk: 0 = any. IN2NAME......Input template map name. Standard defaults. IN2CLASS.....Input template map class. Standard defaults. IN2SEQ.......Input template map seq no. 0 => highest. IN2DISK......Input template map disk: 0 = any. OUTNAME......Output map name. Standard defaults. OUTCLASS.....Output map class. Standard defaults. OUTSEQ.......Output map seq no. 0 => highest unique. OUTDISK......Output map disk. 0 => highest with room. BLC..........The bottom left-hand pixel of the input image which becomes the bottom left corner of the input subimage. The value (0,0) means (1,1). TRC..........The top right-hand pixel of the input image which becomes the top right corner of the subimage. The value (0,0) means take the top right hand corner of the image. IMSIZE.......Output image size in pixels [1=columns, 2=rows] Default is second input image size. The output size is independent of the input size. If IMSIZE is all 0 and APARM(3) and APARM(4) are zero, then the reference pixels are taken from the 2nd image. APARM........Transformation parameters: 1 = Order of Everett interpolation formula [range 1-7: 1=bilinear, 3=bicubic, 5=biquintic, 7=biseptic] 0 => 3. Higher orders will produce more accurate interpolations although oddities may appear when the output cell size is significantly smaller than the input cell size. See extensive discussion in EXPLAIN file. 2 = zero-flag. If > 0.0, zeroes are used wherever blanks would normally be created in the output image. Else "magic value" blanks are used. 3 = X reference pixel in output pixels; 4 = Y reference pixel in output pixels; If BOTH APARM(3) and APARM(4) are zero, but IMSIZE is not, then the numeric center of the input (sub)image is placed at the numeric center of the output image. If IMSIZE is all 0 and both APARM(3) and APARM(4) are zero, then the reference pixels are taken as the reference pixels of the 2nd image.

Explanation of various details of HGEOM (DCW, 11 October 1984) (Modified by W. Cotton Feb, 1993) On the Relation of Input Image Size to Output Image Size: The adverb IMSIZE specifies the dimensions of the output image. If it is set to zero then the default applies and the output image is the same size as the second input. But if it is nonzero it means EXACTLY what it says: the output will have dimensions specified by IMSIZE regardless of what the dimensions of the input image happen to be. Suppose the input image is 800-square and BLC and TRC are zero (i.e., all of the image). If IMSIZE is 100-square then the output is 100-square. On the other hand, if IMSIZE is bigger than the input, say 1000-square, the result is that the input maps to the output and regions which are outside the input image will be blank in the output. Note that the portion of the input which is used is delimited by BLC and TRC completely independently of IMSIZE. The pixel position of the "reference pixel" is specified by the APARM(3) and APARM(4) adverbs. If BOTH are 0 AND IMSIZE is NOT 0, then the input image center (after application of BLC and TRC) is mapped to the numeric center of the output image. If APARM(3), APARM(4), and IMSIZE are all zero, then APARM(3) and APARM(4) are set to the reference pixels of the second image (and IMSIZE is set to the size of the second image). The pixel values given in APARM(3) and APARM(4) are those in the output image. Things like BLC, TRC and IMSIZE are ignored when one or both are non-zero. A Discussion of Blanking in HGEOM: Output pixels are computed as a weighted sum of the input pixels in the neighborhood of the computed location in the input (the interpolation weights depend on the fraction of the pixel spacing). If one of the input pixel values in the sum is blanked, i.e., indefinite in value, then the whole sum, i.e. the output pixel, is blanked. If APARM(2) is true, then instead of blank a zero is delivered as the output pixel value. As a result, if an isolated pixel in the input is blank it will produce blanks (or zeroes) in a region in the output image the same size as the interpolating kernel (4x4 pixels for bicubic, 8x8 for biseptic). This is not nice, but all suggested cures considered so far have other difficulties. There is a special case implemented for the edges of the input image. Here HGEOM "reflects" the pixels near the boundary out into the terra incognita and uses them for interpolation. This means that pixel 1 is copied into the pixel 0 position, pixel 2 is copied into pixel -1, etc. The effect of this algorithm, implemented on all four edges, is that first derivatives of the interpolant are zero at all boundaries. NOTE: "Boundary" in this case also means the window defined by BLC and TRC. If the output image is larger than the transformation of the input window then the extra area will be blanked (or zeroed). Regarding the Everett Interpolation Formula: This program computes where in the input image each output pixel appears. This location is generally at some arbitrary fractional pixel position in x and y, surrounded by four input pixels. The program then interpolates in the pixel values surrounding the position. For APARM(1)=1 it uses linear interpolation, first in x (twice) and then in y. This is called "bilinear" interpolation, and is the default choice. If APARM(1)=3 two pixels on each side in x and y are used, sixteen pixels total. A cubic polynomial is fitted through the four pixels in each of the four rows in x, and evaluated at the computed x-position. Then a cubic is put through those four values (in y) and evaluated. This is "bicubic" interpolation. For APARM(1)=5 we use a 6x6 array, 36 pixels total; this is "biquintic" interpolation. For APARM(1)=7, we have 8x8, 64 pixels, "biseptic" interpolation. Regarding the Numerical Accuracy of Interpolation: The accuracy of interpolation increases with the order of the polynomial, but of course so does the cost of computation. As the polynomial order increases the interpolating function approaches ever more closely to the sinc function, the perfect bandlimited interpolator, as a limit. Bicubic is good enough for many ordinary purposes. Biquintic will probably be good enough even for high dynamic range data like optical CCD imagery and self-calibrated VLA maps. The point is, if we want very accurate results we want the "noise" of interpolation errors to be less than the observational noise in the data at all spatial frequencies of interest. The interpolation is done by computing a set of interpolating weights which are a function of the fractional pixel position where the answer is desired. The answer is then computed by multiplying the weights by the data. This is a convolution. It follows that an interpolator of a given order has a corresponding transfer function. I.e., in the terminology of aperture synthesis, it acts somewhat like a peculiar, probably "bumpy" taper function in the u-v plane. This function is always going to be less than unity in the neighborhood of the Nyquist frequency of the map. The size of this "neighborhood" where the transfer function is significantly different from unity is smaller for higher order polynomials. It follows that maps which are well sampled, say three or four pixels per beam, are easier to interpolate accurately. How to Evaluate the Numerical Errors of Interpolation: DCW is not aware of any proper quantitative treatment of the interpolation accuracy issues anywhere in the literature of image processing. This situation is not as bad as it seems because for critical work there are two simple tests which can be used to estimate the interpolation error: Test1: compute a small change, reverse it, then difference input and output. Use COMB to subtract the second output from the input. The difference is all interpolation noise. If that noise level (due to TWO interpolations) is small enough to suit you then you know that the noise of the first answer is even lower (only one interpolation). Actually, this test is excessively conservative because the transfer function of interpolation is SQUARED by the double calculation (NOTE: any multiple application of interpolation to data will square, cube, etc., the transfer function of data. Mostly one loses high spatial frequencies in the data, but errors creep into the lower spatial frequencies, too.) Test2: compute a small change with one order of interpolation, compute again with increased order, then difference the two results. Use COMB to subtract the the first answer from the second. The difference is an estimate of the error of bicubic interpolation for your image. An even more accurate estimate of the error would be obtained by using APARM(1)=7 (biseptic) for the second computation. Although this test does not inspire the degree of conviction which Test1 can by reproducing the input image, it more than makes up for this by giving an estimate of the interpolation error of a SINGLE operation. NOTE: in performing tests such as are suggested above it will make sense to use BLC and TRC to select a small region of the input image (say, 100-square) which contains interesting structure whose details need to be preserved. This will save time and make it reasonable to perform both Test1 and Test2, and to use biseptic interpolation for Test2. For example, one could perform Test1 using biseptic interpolation in order to be convinced that it's results are accurate. Then perform Test2 for bicubic and/or biquintic and compare them to the biseptic computation.