INNAME Input image (name) INCLASS (class) INSEQ 0.0 9999.0 (seq. #) INDISK 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 2048.0 Output image size in pixels APARM (1)=ref-x (pix),(2)=ref-y, (3)=inclin.ang.,(4)=rot.ang., (5)=border size,(6)=terp.ord, (7)=invert.flag,(8)=spir.flag (9)=spir.rad. ,(10)=zero flag

PGEOM Type: Task Use: PGEOM will transform an input image into polar coordinates and back into rectangular coordinates using Everett interpolation (bilinear, bicubic, or biquintic). PGEOM is capable of doing arbitrary transforms of any size image since it now uses dynamic memory. It can "correct" for inclination and unwrap spiral objects. Be aware that the interpolation method causes edges and blanked pixels to expand into the output image. Adverbs: INNAME........Input map name. Standard defaults. INCLASS... ...Input map class. Standard defaults. INSEQ.........First input map seq no. 0 => highest. INDISK........Input 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. APARM.........Transformation parameters: 1 & 2 = (x, y) location where the "center" is located. This need not be the actual center of the image. It is simply the reference point from which the radius and angle are computed. THERE IS NO DEFAULT. These are the radius zero pixel coords in the input image (APARM(7) <= 0) or the output image (APARM(7) > 0). 3 = inclination angle - the angle between the plane of the object and the plane of the sky (i.e. 0 indicates the object is parallel to the sky, 90 indicates the object is perpendicular to the sky). This parameter allows for deprojection of objects such as elliptical galaxies. 4 = rotation angle - the angle between the plane of the object and a horizontal plane. 5 = border size. This parameter specifies the number of pixels that should be added to each side of the output picture. Thus, negative radii pixels can be created for use by the interpolator. If APARM(5)<=0 it defaults to APARM(6)/2 + 1. 6 = 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. 7 = inversion flag. Causes the inverse of the specified transformation to be performed. Useful to undo previous operation. 8 = spiral flag. Causes the program to unwrap a spiral with a specified reference radius. The reference radius is the radius of the spiral at the rotation angle specified. The program assumes a linear relationship between the angle and radius of the spiral. 9 = spiral radius. This is the reference radius for the spiral. It is the spiral's radius at the specified rotation angle. Must be greater than zero (0.0 => 1.0). 10 = zero-flag. If > 0.0, zeroes are used wherever blanks would normally be created in the output image.

PGEOM - an incomplete explanation. (numerical analysis topics below apply to LGEOM and HGEOM too) On the output headers The output image (when APARM(7) <= 0, i.e. rectangular to polar) has radius on the X axis with increment SQRT (x-incr * y-incr) and angle on the Y axis. If the first two input axes are a sky position pair (i.e. longitude and latitude), the angle is the position angle measured CCW from north. Maintaining header information in this transformation is tricky. Most AIPS programs will not give correct RA and Decs for pixels in the radius-angle output image. Header information is not lost however so long as the input rectangular image has no more than 5 axes. If your header has more than 5, use TRANS to make sure that axes 6 and 7 are null (only 1 pixel). On the Relation of Input Image Size to Output Image Size: The adverb IMSIZE specifies the dimensions of the output image more or less. When APARM(7) <= 0 (rectangualr to polar), the number of rows is IMSIZE(2) + 2*APARM(5) with a default for IMSIZE(2) of TRC(2)-BLC(2)+1. The number of columns is set to the maximum of IMSIZE(1) and some side of the input subimage. If IMSIZE(1) <= 0, the number of columns is the diagonal of the input image in pixels. One APARM(5) is then added. When APARM(7) > 0 (polar to rectangular), the number of rows is IMSIZE(2) - 2*APARM(5) with the same default. The number of columns is IMSIZE(1) with a default of 2 * (TRC(1) - BLC(1) + 1 -APARM(5)) (note the 2). A Discussion of Blanking in PGEOM: 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(9) 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 PGEOM "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(6)=1 it uses linear interpolation, first in in x (twice) and then in y. This is called "bilinear" interpolation, and is the default choice. If APARM(6)=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(6)=5 we use a 6x6 array, 36 pixels total; this is "biquintic" interpolation. For APARM(6)=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 these accuracy issues anywhere in the literature. 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 simple change, reverse it, then difference input and output. E.g., transform the input image into polar coordinates, then invert the transformation. 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, the rotated image, 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. E.g., transform the input image into polar coordinates with APARM(6)=3 (bicubic), then perform the operation again with APARM(6)=5 (biquintic). 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(6)=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. See EXPLAIN LGEOM for a discussion of Virtual Memory in PGEOM