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Spectral Extraction of Extended Sources using Wavelet Interpolation Paul Barrett Linda Dressel STIS Calibration Group 2005 July 21 Raw Spectral Image Interpolated Spectral Image Interpolating Subdivision Construct a polynomial p of degree N-1: p(xj,k+n) = yj,k+n for -D < nD Calculate value at midpoint: yj+1,2k+1 = p(xj+1,2k+1) Average Interpolation Subdivision Construct a polynomial p of degree N-1: int p(x)dx = yj,k+n for D<n<D Calculate two coeff. at k and k+0.5 Note: the polynomial fits the cumulative values. Wavelet Properties Compact Support: w(x) is exactly zero outside the interval [-N+1, N]. Average-interpolation: w(x) is averageinterpolating in the sense that int k+1 k f(x)dx = yk,0. Symmetry: w(x) is symmetric about x = . Polynomial reproduction: w(x) reproduces polynomials up to degree N-1. Wavelet Properties Smoothness: R(x) is continuous of order R, with R = R(N) > 0. Refinability: (x) satisfies a refinement relation of the form (x) = Nl=-N+1hl (2x-l). The construction implies that h0 = h1 = 1 and h2 = h2l+1 if l0. This last property distinguishes wavelets from filter functions. Average Interpolation Algorithm Step 1: Subdivide pixel into 2 subpixels using an N-order (=7) polynomial to partition the counts in. Step 2: Apply inverse Haar transform (wavelet). Step 3: Repeat j times. Step 4: Convolve subpixels using instrumental point spread function (PSF) Spectral Extraction Future Work Develop a 1 step algorithm for the interpolation by finding a wavelet that approximates the detector PSF. – This will remove the convolution step which slightly degrades the spectral resolution. Remove the sawtooth pattern from the off-trace extractions by interpolating the border subpixels. Calculate the errors for each subpixel.