diff --git a/wavelet2.h b/wavelet2.h index 7c57c03..deabc5e 100644 --- a/wavelet2.h +++ b/wavelet2.h @@ -1,246 +1,243 @@ /* ========================================================================= Definitions for 2D wavelet transforms on arrays of 3-tuples. Roland Schregle (roland.schregle@{hslu.ch, gmail.com}) (c) Lucerne University of Applied Sciences and Arts, supported by the Swiss National Science Foundation (SNSF #179067, "Light Fields for Spatio-Temporal Glare Assessment") ========================================================================= $Id$ */ #ifndef _WAVELET2_H #define _WAVELET2_H /* Wavelet coefficient type; defaults to double if not already defined. (may be overridden in compiler command line, for example). NOTE: single precision float may improve cache performance during 2D transform with large arrays. */ #ifndef WAVELET_COEFF #define WAVELET_COEFF double #endif /* Perform final Haar wavelet transform on coarsest two detail levels. * This has the advantage of resulting in a single coarsest level * approximation coefficient, possibly at the expense of poorer * compression */ #define WAVELET_FINAL_HAAR /* Boundary extension modes for padded Daubechies D4 wavelet transform */ #define WAVELET_EXTEND_CONST 0 /* Constant (default if undef) */ #define WAVELET_EXTEND_GRAD1 1 /* 1st order gradient */ #define WAVELET_EXTEND_GRAD2 2 /* 2nd order gradient */ - #if 0 - /* Not implemented (yet) */ - #define WAVELET_EXTEND_PERIOD 3 /* Periodic (wraparound) */ - #define WAVELET_EXTEND_SYMM 4 /* Symmetric (reflection) */ - #endif + #define WAVELET_EXTEND_CYCL 3 /* Periodic (wraparound) */ + #define WAVELET_EXTEND_SYMM 4 /* Symmetric (reflection) */ #ifndef WAVELET_EXTEND - #define WAVELET_EXTEND_MODE WAVELET_EXTEND_GRAD2 + #define WAVELET_EXTEND_MODE WAVELET_EXTEND_GRAD1 #endif /* Convenience macros for handling coefficients */ #define min(a, b) ((a) < (b) ? (a) : (b)) #define max(a, b) ((a) > (b) ? (a) : (b)) #define coeffAvg(a) (((a) [0] + (a) [1] + (a) [2]) / 3) #define coeffIsEmpty(c) ( \ isnan((c) [0]) || isnan((c) [1]) || isnan((c) [2]) \ ) #define coeffIsZero(c) ((c) [0] == 0 && (c) [1] == 0 && (c) [2] == 0) /* Determine if coeff tuple c is thresholded, also including empty coeffs * in debugging mode, since these will be unconditionally dropped. Note * that the coarsest approximation coefficients are not thresholded, * since they are essential for reconstruction! */ #ifdef WAVELET_FINAL_HAAR #define coeffThresh(c, i, j, t) ( \ fabs(t) > 0 && ((i) || (j)) && ( \ coeffIsEmpty((c) [i][j]) || fabs((c) [i][j][0]) < (t) || \ fabs((c) [i][j][1]) < (t) || fabs((c) [i][j][2]) < (t) \ ) \ ) #else #define coeffThresh(c, i, j, t) ( \ fabs(t) > 0 && ((i) > 2 || (j) > 2) && ( \ coeffIsEmpty((c) [i][j]) || fabs((c) [i][j][0]) < (t) || \ fabs((c) [i][j][1]) < (t) || fabs((c) [i][j][2]) < (t) \ ) \ ) #endif /* Haar / Daubechies D4 coefficients */ #define SQRT2 1.41421356 #define SQRT3 1.73205081 #define H4NORM (0.25 / SQRT2) /* Haar wavelet coeffs */ extern const WAVELET_COEFF h2; /* Daubechies D4 wavelet coeffs */ extern const WAVELET_COEFF h4 [4]; extern const WAVELET_COEFF g4 [4]; /* Wavelet matrix defs; these are stored as arrays of pointers (a.k.a. * Iliffe vectors), as this proved to be more performant than a flattened * representation. Encapsulating the coefficient's triplets in a struct * prevents the compiler from introducing another layer of indirection. * */ typedef WAVELET_COEFF WaveletCoeff3 [3]; #define WAVELET_COEFFSIZE (sizeof(WaveletCoeff3)) typedef WaveletCoeff3 **WaveletMatrix2; /* ----------- 2D COEFFICIENT MATRIX ALLOC/INIT ROUTINES ----------- */ WaveletMatrix2 allocWaveletMatrix2 (unsigned len); /* Allocate and return a 2D coefficient array of size (len x len). Returns NULL if allocation failed. */ void freeWaveletMatrix2 (WaveletMatrix2 y, unsigned len); /* Free previously allocated 2D coeff array y of size (len x len) */ void zeroCoeffs2 (WaveletMatrix2 y, unsigned len); /* Set 2D array coefficients to zero */ /* ---------- 2D WAVELET TRANSFORM OPTIMISED FOR SIZES 2^l ---------- ------ THESE USE PERIODIC (WRAP-AROUND) BOUNDARY EXTENSION ------- */ int waveletXform2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned l); /* Perform full 2D multiresolution forward wavelet transform on array y of size (2^l) x (2^l) containing original signal as 3-tuples, where l >= 1. Note no intra-tuple transform occurs. The wavelet coefficients are returned in array y, containing the coarsest approximation in y [0][0] followed by horizontal/vertical details in order of increasing resolution/frequency. A preallocated array yt of identical dimensions to y can be supplied as buffer for intermediate results. If yt == NULL, a buffer is automatically allocated and freed on demand, but this is inefficient for frequent calls. It is recommended to preallocate yt to the maximum expected size. The dimensions of yt are not checked; this is the caller's responsibility. Returns 0 on success, else -1. */ int waveletInvXform2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned l); /* Perform full 2D multiresolution inverse wavelet transform on array y of size (2^l) x (2^l) containing wavelet coefficients as 3-tuples, where l >= 1. Note no intra-tuple transform occurs. A preallocated array yt of identical dimensions to y can be supplied as buffer for intermediate results. If yt == NULL, a buffer is automatically allocated and freed on demand, but this is inefficient for frequent calls. It is recommended to preallocate yt to the maximum expected size. The dimensions of yt are not checked; this is the caller's responsibility. The reconstructed signal is returned in array y. Returns 0 on success, else -1. */ /* ------------ 2D WAVELET TRANSFORM FOR ARBITRARY SIZES ------------- ---- THESE USE CONSTANT BOUNDARY EXTENSION WITH PADDING COEFFS----- */ unsigned padWaveletXform2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned yLen0, unsigned *numNonZero ); /* Perform full 2D multiresolution forward wavelet transform with padded boundary coefficients on array y containing (yLen0 x yLen0) samples of 3-tuples. Note no intra-tuple transform occurs. An additional array yt of identical dimension to y is required as buffer for intermediate results. The wavelet coefficients are returned in array y, containing the coarsest approximation coefficients in y [0][0] followed by horizontal/vertical detail coefficients in order of increasing resolution/frequency. NOTE: Both y and yt must be large enough to accommodate the additional padding coefficients introduced at the array boundaries during the transform. It is the caller's responsibility to preallocate and dimension these arrays accordingly with allocWaveletMatrix2(). The dimension of the output array (which generally exceeds the input dimension yLen0) must be interrogated beforehand by calling the function with y or yt set to NULL. The returned value is the output dimension, or 0 if the transform failed. In addition, the number of nonzero coefficients is returned in numNonZero (if not NULL), which is a subset of the number of output coefficients, i.e. (output dimension)^2. */ unsigned padWaveletInvXform2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned yLen0, unsigned yLenOrg ); /* Perform full 2D multiresolution inverse wavelet transform on array y containing (yLen0 x yLen0) boundary padded wavelet coefficients as 3-tuples. The original number of samples (yLenOrg x yLenOrg) is required for the reconstruction. An additional array yt of identical dimension to y is required as buffer for intermediate results. The reconstructed samples are returned in y. This is smaller than (yLen0 x yLen0) since the boundary coefficients from the forward transform are collapsed. The returned value is the reconstructed number of samples per dimension (yLenOrg), or 0 if the inverse transform failed. */ #ifdef WAVELET_DBG /* ----------------- SUPPORT ROUTINES FOR DEBUGGING ----------------- */ void clearCoeffs2 (WaveletMatrix2 y, unsigned len); /* Clear 2D array to facilitate debugging */ void dumpCoeffs2 (const WaveletMatrix2 y1, const WaveletMatrix2 y2, unsigned y1Len, unsigned y2Len, float thresh ); /* Dump 2D arrays y1 and y2 of dims y1Len resp. y2Len side-by-side to stdout (skip y2 if NULL). Coefficients with absolute magnitude less than thresh are marked with brackets ('[]') as thresholded. */ float rmseCoeffs2 (const WaveletMatrix2 y1, const WaveletMatrix2 y2, unsigned len ); /* Calculate RMSE between 2D matrices y1 and y2 */ #endif #endif diff --git a/wavelet3.c b/wavelet3.c index 40a6130..0cfec25 100644 --- a/wavelet3.c +++ b/wavelet3.c @@ -1,1049 +1,1078 @@ /* ========================================================================= 2-dimensional Daubechies D4 and Haar wavelet transforms on arbitrary sized arrays of 3-tuples. Array boundaries are extended as defined by WAVELET_EXTEND_MODE, which introduces padding coefficients. Compile with -DWAVELET_TEST_2DPAD to build standalone unit tests. The implementation here is mainly distilled from the following sources: -- Filip Wasilewski's pywavelets: https://github.com/PyWavelets/pywt/tree/master/pywt/_extensions/c [D4 with boundary condition handling via padding, arbitrary sizes] -- Ingrid Daubechies and Wim Sweldens, "Factoring Wavelet Tranforms into Lifting Steps", section 7.5: https://services.math.duke.edu/~ingrid/publications/J_Four_Anal_Appl_4_p247.pdf [Lifting scheme on D4] -- Ian Kaplan's webpages on wavelets and signal processing:: http://bearcave.com/misl/misl_tech/wavelets/daubechies/index.html http://bearcave.com/misl/misl_tech/wavelets/lifting/basiclift.html [Lifting scheme on D4] -- Stefan Moebius' TurboWavelets.Net: https://github.com/codeprof/TurboWavelets.Net/tree/master/TurboWavelets [Boundary condition handling with arbitrary sizes, backreferencing] -- "Numerical Recipes in C", chapter 13.10: http://cacs.usc.edu/education/cs653/NRC-c13.10-Wavelets.pdf [Linear algebra (matrix mult.) implementation of D4] Roland Schregle (roland.schregle@{hslu.ch, gmail.com}) (c) Lucerne University of Applied Sciences and Arts, supported by the Swiss National Science Foundation (SNSF #179067, "Light Fields for Spatio-Temporal Glare Assessment") ========================================================================= $Id$ */ #include "wavelet2.h" #include #include #include #include static unsigned padHaarStep2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned len ) /* Single iteration of forward 2D Haar wavelet transform with padding on array y of arbitrary size (len x len) containing 3-tuples. This variant uses constant boundary extension. The transform is performed over y's 2nd axis (i.e. horizontally, assuming row-major addressing as per C convention). Note the triplets per array element are _not_ decorrelated, but transformed independently. The wavelet coefficients are returned in the *TRANSPOSED* output array yt, of identical dimensions to y. The transpose arranges the generated coefficients vertically, and prepares the array for another transform over its 2nd axis in a subsequent call (this time as input y). The result of a subsequent transform restores yt to y's original orientation, in which case both horizontal and vertical axes have been decorrelated. Returns array dimension len on success, else 0. */ { static unsigned axis = 0; const unsigned hLen = -(-(signed)len >> 1); int h, i, j, k; if (len < 2 || !y || !yt) /* Input shorter than wavelet support, or no input/output */ return 0; #ifdef WAVELET_DBG clearCoeffs2(yt, len); #endif /* NOTE: yt is transposed on the fly such that the next function call * transforms over the alternate axis. This is done by simply swapping * the indices during assignment */ for (i = 0; i < len; i++) { for (j = 0; j <= len - 2; j += 2) { h = j >> 1; for (k = 0; k < 3; k++) { /* Smooth/approx/avg/lowpass */ yt [h ] [i] [k] = h2 * ( y [i] [j ] [k] + y [i] [j + 1] [k] ); /* Detail/diff/highpass */ yt [h + hLen] [i] [k] = h2 * ( y [i] [j ] [k] - y [i] [j + 1] [k] ); } } if (len & 1) { /* Odd length; pad with last value as additional approx term */ for (k = 0; k < 3; k++) /* Smooth/approx/avg/lowpass */ yt [hLen - 1] [i] [k] = y [i] [len - 1] [k]; } } #ifdef WAVELET_DBG printf("%s FWD HAAR (%d x %d)\n", axis ? "VERT" : "HORIZ", len, len); dumpCoeffs2(y, yt, len, len, 0); putchar('\n'); #endif axis ^= 1; return len; } static unsigned padHaarInvStep2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned len ) /* Single iteration of inverse 2D Haar wavelet transform with padding on array y of arbitrary size (len x len) containing 3-tuples. This reverses the forward transform above. The transform is inverted over y's 2nd axis (i.e. horizontally, assuming row-major addressing as per C convention). The inverted coefficients are returned in the *TRANSPOSED* output array yt, of identical dimensions to y. The transpose arranges the inverted coefficients vertically, and prepares the array for another inverse transform over its 2nd axis in a subsequent call (this time as input y). The result of a subsequent inverse transform restores yt to y's original orientation, in which case both horizontal and vertical axes have been inversely transformed. Returns array dimension len on success, else 0. */ { static unsigned axis = 1; const unsigned hLen = -(-(signed)len >> 1); int h, i, j, k; if (len < 2 || !y || !yt) /* Too few coeffs for reconstruction, or no input/output */ return 0; #ifdef WAVELET_DBG clearCoeffs2(yt, len); #endif /* NOTE: i, j are swapped relative to the forward transform, as axis * order is now reversed. */ /* NOTE: yt is transposed on the fly such that the next function call * inverts over the alternate axis. This is done by simply swapping * the indices during assignment */ for (i = 0; i < len; i++) { for (j = 0; j <= len - 2; j += 2) { h = j >> 1; for (k = 0; k < 3; k++) { yt [i] [j ] [k] = h2 * ( y [h ] [i] [k] + /* Approx */ y [h + hLen] [i] [k] /* Detail */ ); yt [i] [j + 1] [k] = h2 * ( y [h ] [i] [k] - /* Approx */ y [h + hLen] [i] [k] /* Detail */ ); } } if (len & 1) { /* Odd length; pad additional value from last approximation term */ for (k = 0; k < 3; k++) yt [i] [len - 1] [k] = y [hLen - 1] [i] [k]; } } #ifdef WAVELET_DBG printf("%s INV HAAR (%d x %d)\n", axis ? "VERT" : "HORIZ", len, len); dumpCoeffs2(y, yt, len, len, 0); putchar('\n'); #endif axis ^= 1; return len; } static unsigned padD4Step2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned yLen, unsigned ytOffset ) /* Single iteration of forward 2D Daubechies D4 wavelet transform with boundary padding on array y containing (yLen x yLen) 3-tuples. This variant extends the input array beyond its boundary as defined by WAVELET_EXTEND_MODE, which results in padding coefficients. The transform is performed over y's 2nd axis (i.e. horizontally, assuming row-major addressing as per C convention). Note the triplets per array element are _not_ decorrelated, but transformed independently. The output wavelet coefficients are returned in the *TRANSPOSED* array yt. The transpose arranges the generated coefficients vertically, and prepares the array for another transform over its 2nd axis in a subsequent call (this time as input y). The result of a subsequent transform restores yt to y's original orientation, in which case both horizontal and vertical axes have been decorrelated. Both the input array y and output array yt must be large enough to also accommodate the additional padding coefficients introduced at the boundaries. The number of output coefficient pairs ytLen returned in yt is obtained as return value, and can be interrogated beforehand without performing a transform by passing either y or yt as NULL. The approximation coefficients are returned in yt [0..ytLen-1] [i], while the details are returned in yt [ytOffset..ytOffset+ytLen-1] [i]. The offset ytOffset must be supplied by the caller to accommodate the detail coefficients (plus padding!) without clobbering the approximation coefficients (again, plus padding!). Consequently, there is a gap between the approximation coefficients in the upper left of the matrix, and the detail coefficients in the lower right. This gap shrinks and shifts toward the upper left with each iteration. The following schema indicates the matrix contents during the transform (see the unit test output when compiled with WAVELET_TEST_2DPAD): ----> j +----------+--------------+ +-----------+-------------+ | | y_ij | | | S(y_ji) | | | +----------+ | (H_k xform)^T +-----------+ | v | | ----------> | | i | | +-----------+ | | | | D(y_ji) | | +-------------------------+ +-------------------------+ ----> i +----------+--------------+ +----------+---+----------+ | | S(y_ji) | | |S(S(y_ij))| |S(D(y_ij))| | +----------+ | (V_k xform)^T +----------+ +----------+ v | | ----------> | | j +----------+ | +----------+ +----------+ | D(y_ji) | | |D(S(y_ij))| |D(D(y_ij))| +-------------------------+ +----------+---+----------+ ... ... ... ... ... ... +------+------+--------+--------+----------------+ |SS_n-1|SD_n-1| | | | |------+------+ SD_n-2 | | | +DS_n-1|DD_n-1+--------+ . . . | SD_0 | +------+----+-+--------+ | | | DS_n-2 | | DD_n-2 | +----------------+ (V_n-1 xform)^T +-----------+ +--------+ . . . | ----------> | . . . . . | | . . . . . | +----------------+ +----------------+ | | | | | DS_0 | . . . . . . | DD_0 | | | | | | | | | +----------------+--------------+----------------+ where i, j are the array indices of the outer resp. inner loop, S(y_ij) D(y_ij) are approximation resp. details coefficients of input y_ij (itself either the original signal or approximation/detail coefficients from previous iterations), and H_k, V_k are the horizontal resp. vertical transforms of the k-th (out of the maximum n) iteration. */ { static unsigned axis = 0; - unsigned h, i, j, k, l, i0 = 0, iN, wIdx; + unsigned h, i, j, k, l, i0 = 0, iN, wIdx, dExt; /* Number of output coeff pairs ytLen, incl. boundary padding, and max dimension of entire array */ const unsigned ytLen = (yLen + 3) >> 1, ytTotalLen = ytOffset + ytLen; + WAVELET_COEFF yExt; #if (WAVELET_EXTEND_MODE == WAVELET_EXTEND_GRAD1 || \ WAVELET_EXTEND_MODE == WAVELET_EXTEND_GRAD2 \ -) +) /* Gradient boundary extension stuff */ - unsigned dExt; - WAVELET_COEFF yExt, dy1_0, dy1_1, dy2_0; + WAVELET_COEFF dy1_0, dy1_1, dy2_0; #endif if (y && yt && yLen >= 2) { #ifdef WAVELET_DBG /* Clear output array yt to facilitate debugging */ clearCoeffs2(yt, ytTotalLen); #else /* Zero output (sub)array yt; note this will not affect accumulated * coefficients from previous iterations, as they lie outside the * array bounds */ zeroCoeffs2(yt, ytTotalLen); #endif /* FWD TRANSFORM GOTCHAS: * * Each full iteration of the forward transform consists of a * horizontal step followed by a vertical one. yt is transposed on * the fly such that the next function call transforms over the * alternate axis. This is done by simply swapping the indices during * assignment. * * In the initial horizontal transform, the input array only contains * a set of approximation coefficients (or the original signal) at * y [0..yLen-1] [0..yLen-1]. * * In the subsequent vertical transform, the input array then contains * two sets of coefficients from the preceding horizontal transform: * approximation coeffs y [0..ytLen-1] [0..ytLen-1] and detail coeffs * y [ytOffset..ytTotalLen-1] [ytOffset..ytTotalLen-1]. In this case, * the loop below is iterated twice with an offset yOffset to select * both sets of coefficients to transform. Note the detail coeffs are * selected first to facilitate the looping logic; if i0==0 at the end * of the loop (i.e. approx coeffs were selected), it terminates. */ do { i0 ^= ytOffset * axis; iN = i0 ? ytTotalLen : yLen; for (i = i0; i < iN; i++) { /* First iteration (j = -2); pad coefficients at left edge and * extend input beyond boundary according to extension mode */ for (k = 0; k < 3; k++) { #if (WAVELET_EXTEND_MODE == WAVELET_EXTEND_CONST || \ !defined(WAVELET_EXTEND_MODE) \ ) /* Constant boundary extension (=default) */ /* Smooth/approx/avg/lowpass */ yt [0 ] [i] [k] = (h4 [0] + h4 [1] + h4 [2]) * y [i] [0] [k] + (h4 [3] ) * y [i] [1] [k]; /* Detail/diff/highpass */ yt [ytOffset] [i] [k] = (g4 [0] + g4 [1] + g4 [2]) * y [i] [0] [k] + (g4 [3] ) * y [i] [1] [k]; - + #elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_GRAD1 /* 1st order gradient boundary extension */ dy1_0 = y [i] [0] [k] - y [i] [1] [k]; /* Smooth/approx/avg/lowpass */ yt [0 ] [i] [k] = h4 [0] * (y [i] [0] [k] + 2 * dy1_0) + h4 [1] * (y [i] [0] [k] + dy1_0) + h4 [2] * (y [i] [0] [k] ) + h4 [3] * (y [i] [1] [k] ); /* Detail/diff/highpass */ yt [ytOffset] [i] [k] = g4 [0] * (y [i] [0] [k] + 2 * dy1_0) + g4 [1] * (y [i] [0] [k] + dy1_0) + g4 [2] * (y [i] [0] [k] ) + g4 [3] * (y [i] [1] [k] ); #elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_GRAD2 /* 2nd order gradient boundary extension */ dy1_0 = y [i] [0] [k] - y [i] [1] [k]; dy1_1 = y [i] [1] [k] - y [i] [2] [k]; dy2_0 = dy1_0 - dy1_1; /* Smooth/approx/avg/lowpass */ yt [0 ] [i] [k] = /* h4 [0] * (y [i] [0] [k] + 2 * (dy1_0 + 1 * dy2_0)) + h4 [1] * (y [i] [0] [k] + 1 * (dy1_0 + 0 * dy2_0)) + h4 [2] * (y [i] [0] [k] + 0 * (dy1_0 )) + h4 [3] * (y [i] [1] [k] ); */ h4 [0] * (y [i] [0] [k] + 2 * (dy1_0 + dy2_0)) + h4 [1] * (y [i] [0] [k] + dy1_0 ) + h4 [2] * (y [i] [0] [k] ) + h4 [3] * (y [i] [1] [k] ); /* Detail/diff/highpass */ yt [ytOffset] [i] [k] = /* g4 [0] * (y [i] [0] [k] + 2 * (dy1_0 + 1 * dy2_0)) + g4 [1] * (y [i] [0] [k] + 1 * (dy1_0 + 0 * dy2_0)) + g4 [2] * (y [i] [0] [k] + 0 * (dy1_0 )) + g4 [3] * (y [i] [1] [k] ); */ g4 [0] * (y [i] [0] [k] + 2 * (dy1_0 + dy2_0)) + g4 [1] * (y [i] [0] [k] + dy1_0 ) + g4 [2] * (y [i] [0] [k] ) + g4 [3] * (y [i] [1] [k] ); + +#elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_CYCL + /* Cyclic (perdiodic / wraparound) boundary extension */ + /* Smooth/approx/avg/lowpass */ + yt [0 ] [i] [k] = + h4 [0] * y [i] [yLen - 2] [k] + + h4 [1] * y [i] [yLen - 1] [k] + + h4 [2] * y [i] [0 ] [k] + + h4 [3] * y [i] [1 ] [k]; + + /* Detail/diff/highpass */ + yt [ytOffset] [i] [k] = + g4 [0] * y [i] [yLen - 2] [k] + + g4 [1] * y [i] [yLen - 1] [k] + + g4 [0] * y [i] [0 ] [k] + + g4 [3] * y [i] [1 ] [k]; + +#elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_SYMM + /* Symmetric (reflected) boundary extension */ + /* Smooth/approx/avg/lowpass */ + yt [0 ] [i] [k] = + h4 [0] * y [i] [2] [k] + + h4 [1] * y [i] [1] [k] + + h4 [2] * y [i] [0] [k] + + h4 [3] * y [i] [1] [k]; + + /* Detail/diff/highpass */ + yt [ytOffset] [i] [k] = + g4 [0] * y [i] [2] [k] + + g4 [1] * y [i] [1] [k] + + g4 [2] * y [i] [0] [k] + + g4 [3] * y [i] [1] [k]; #endif } /* Intermediate and last iterations (j >= 0) */ for (j = 0; j < yLen; j += 2) { h = (j >> 1) + 1; for (k = 0; k < 3; k++) { #ifdef WAVELET_DBG /* Only necessary in debugging mode, since otherwise already cleared by zeroCoeffs2() */ yt [h] [i] [k] = yt [h + ytOffset] [i] [k] = 0; #endif /* Convolve up to boundary of wavelet or input signal, * whichever comes first */ for (l = j; l < min(j + 4, yLen); l++) { /* Smooth/approx/avg/lowpass */ yt [h ] [i] [k] += h4 [l - j] * y [i] [l] [k]; /* Detail/diff/highpass */ yt [h + ytOffset] [i] [k] += g4 [l - j] * y [i] [l] [k]; } /* If necessary, convolve beyond input signal boundary * with extension. This is skipped if the wavelet boundary * was already reached */ for (; l < j + 4; l++) { wIdx = l - j; #if (WAVELET_EXTEND_MODE == WAVELET_EXTEND_CONST || \ !defined(WAVELET_EXTEND_MODE) \ ) /* Constant boundary extension (=default) */ - /* Smooth/approx/avg/lowpass */ - yt [h ] [i] [k] += h4 [wIdx] * - y [i] [yLen - 1] [k]; - - /* Detail/diff/highpass */ - yt [h + ytOffset] [i] [k] += g4 [wIdx] * - y [i] [yLen - 1] [k]; + yExt = y [i] [yLen - 1] [k]; #elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_GRAD1 /* 1st order gradient boundary extension */ dy1_0 = y [i] [yLen - 1] [k] - y [i] [yLen - 2] [k]; dExt = l - yLen + 1; yExt = y [i] [yLen - 1] [k] + dExt * dy1_0; - /* Smooth/approx/avg/lowpass */ - yt [h ] [i] [k] += h4 [wIdx] * yExt; - - /* Detail/diff/highpass */ - yt [h + ytOffset] [i] [k] += g4 [wIdx] * yExt; - #elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_GRAD2 /* 2nd order gradient boundary extension */ dy1_0 = y [i] [yLen - 1] [k] - y [i] [yLen - 2] [k]; dy1_1 = y [i] [yLen - 2] [k] - y [i] [yLen - 3] [k]; dy2_0 = dy1_0 - dy1_1; dExt = l - yLen + 1; yExt = y [i] [yLen - 1] [k] + dExt * (dy1_0 + (dExt ? dExt - 1 : 0) * dy2_0); - + +#elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_CYCL + /* Cyclic (perdiodic/wraparound) boundary extension */ + yExt = y [i] [l - yLen] [k]; + +#elif WAVELET_EXTEND_MODE == WAVELET_EXTEND_SYMM + /* Symmetric (reflected) boundary extension */ + yExt = y [i] [((yLen - 1) << 1) - l] [k]; + /* = (yLen - 1) - (l - (yLen - 1)) */ +#endif + /* Smooth/approx/avg/lowpass */ yt [h ] [i] [k] += h4 [wIdx] * yExt; - + /* Detail/diff/highpass */ yt [h + ytOffset] [i] [k] += g4 [wIdx] * yExt; -#endif } } } } } while (i0); #ifdef WAVELET_DBG if (axis) printf("VERT FWD D4: (%dx%d) H-approx, (%dx%d) H-detail\n" "-> (%dx%d) V-approx of H-approx, (%dx%d) V-approx of H-detail,\n" " (%dx%d) V-detail of H-approx, (%dx%d) V-detail of V-detail\n", ytLen, yLen, ytLen, yLen, ytLen, ytLen, ytLen, ytLen, ytLen, ytLen, ytLen, ytLen ); else printf( "HORIZ FWD D4: (%dx%d) -> (%dx%d) H-approx, (%dx%d) H-detail\n", yLen, yLen, ytLen, yLen, ytLen, yLen ); dumpCoeffs2(y, yt, ytTotalLen, ytTotalLen, 0); putchar('\n'); #endif axis ^= 1; } return ytLen; } static unsigned padD4InvStep2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned yLen, unsigned yOffset ) /* Single iteration of inverse 2D Daubechies D4 wavelet transform with boundary padding on array y containing pairs of (yLen x yLen) approximation and detail coefficients in 3-tuples. This reverses the forward transform above. The transform is inverted over y's 2nd axis (i.e. horizontally, assuming row-major addressing as per C convention). The inverted coefficients are returned in the *TRANSPOSED* output array yt. The transpose arranges the inverted coefficients vertically, and prepares the array for another inverse transform over its 2nd axis in a subsequent call (this time as input y). The result of a subsequent inverse transform restores yt to y's original orientation, in which case both horizontal and vertical axes have been inversely transformed. The number of output coefficients is smaller than the input as padding coefficients are collapsed at the boundaries during the inverse transform. The number of output coefficients per dimension ytLen is obtained as returned value, and can be interrogated beforehand without performing a transform by passing either y or yt as NULL. The inversion expects approximation coeffs y [0..yLen-1][0..yLen-1] and y [0..yLen-1] [yOffset..yOffset+yLen-1], and detail coefficients y [yOffset..yOffset+yLen-1] [0..yLen-1] and y [yOffset..yOffset+yLen-1] [yOffset..yOffset+yLen-1]. The offset yOffset must be supplied by the caller to account for the fact that the number of output coefficients ytLen is smaller than yLen due to the collapsed padding coefficients. Consequently there is a gap between the inverted coefficients located in the corners of the matrix. This gap increases with each iteration, until all coefficients have been consumed by the inversion. */ { static unsigned axis = 1; unsigned h, i, j, k, l, i0 = 0, iN; /* Number of output coeff (pairs) ytLen, max dimension of entire array */ unsigned ytLen = (yLen - 1) << 1; const unsigned yTotalLen = yOffset + yLen; if (y && yt && ytLen >= 2) { #ifdef WAVELET_DBG /* Clear output array yt to facilitate debugging */ clearCoeffs2(yt, yTotalLen); #else /* Zero output (sub)array yt; note this will not affect accumulated * coefficients from previous iterations, as they lie outside the * array bounds */ zeroCoeffs2(yt, yTotalLen); #endif /* INV TRANSFORM GOTCHAS: * * i, j are swapped relative to the forward transform, as axis order * is now reversed (vertical before horizontal xform). yt is * transposed on the fly such that the next function call transforms * over the alternate axis. This is done by simply swapping the * indices during assignment. * * In the inverse vertical transform, the input array contains two * sets of coefficients: * (a) approx of approx coeffs y [0..yLen-1] [0..yLen-1] in the * upper left, and details of approximation coefficients * y [yOffset..yTotalLen-1] [0..yLen-1] in the lower left. * (b) approx of detail coeffs y [0..yLen-1] [yOffset..yTotalLen-1] * in the upper right, and details of detail coefficients * y [yOffset..yTotalLen-1] [yOffset..yTotalLen-1] in lower right. * * The loop below is iterated twice with an offset to select both sets * of coefficients to inverse the transform. Note set (b) is selected * first to facilitate the looping logic; if i0==0 at the end of the * loop (i.e. set (a) was selected), it terminates. */ do { i0 ^= yOffset * (axis); iN = i0 ? yTotalLen : ytLen; for (i = i0; i < iN; i++) { for (j = 0; j < ytLen; j += 2) { h = j >> 1; for (k = 0; k < 3; k++) { yt [i] [j ] [k] = h4 [2] * y [h ] [i] [k] + /* Approx */ g4 [2] * y [h + yOffset ] [i] [k] + /* Detail */ h4 [0] * y [h + 1 ] [i] [k] + /* Approx */ g4 [0] * y [h + 1 + yOffset] [i] [k]; /* Detail */ yt [i] [j + 1] [k] = h4 [3] * y [h ] [i] [k] + /* Approx */ g4 [3] * y [h + yOffset ] [i] [k] + /* Detail */ h4 [1] * y [h + 1 ] [i] [k] + /* Approx */ g4 [1] * y [h + 1 + yOffset] [i] [k]; /* Detail */ } } } } while (i0); #ifdef WAVELET_DBG if (axis) printf("VERT INV D4: " "(%dx%d) V-approx of H-approx, (%dx%d) V-approx of H-detail,\n" "(%dx%d) V-detail of H-approx, (%dx%d) V-detail of V-detail\n" "-> (%dx%d) H-approx, (%dx%d) H-detail\n", yLen, yLen, yLen, yLen, yLen, yLen, yLen, yLen, yLen, ytLen, yLen, ytLen ); else printf( "HORIZ INV D4: (%dx%d) H-approx, (%dx%d) H-detail -> (%dx%d)\n", yLen, ytLen, yLen, ytLen, ytLen, ytLen ); dumpCoeffs2(y, yt, yTotalLen, yTotalLen, 0); putchar('\n'); #endif axis ^= 1; } return ytLen; } unsigned padWaveletXform2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned yLen0, unsigned *numNonZero ) /* Perform full 2D multiresolution forward wavelet transform with padded boundary coefficients on array y containing (yLen0 x yLen0) samples of 3-tuples. Note no intra-tuple transform occurs. An additional array yt of identical dimension to y is required as buffer for intermediate results. The wavelet coefficients are returned in array y, containing the coarsest approximation coefficients in y [0][0] followed by horizontal/vertical detail coefficients in order of increasing resolution/frequency. NOTE: Both y and yt must be large enough to accommodate the additional padding coefficients introduced at the array boundaries during the transform. It is the caller's responsibility to preallocate and dimension these arrays accordingly with allocWaveletMatrix2(). The dimension of the output array (which generally exceeds the input dimension yLen0) must be interrogated beforehand by calling the function with y or yt set to NULL. The returned value is the output dimension, or 0 if the transform failed. In addition, the number of nonzero coefficients is returned in numNonZero (if not NULL), which is a subset of the number of output coefficients, i.e. (output dimension)^2. */ { /* Output size (number of coeffs, including boundary padding coeffs), output array offset and pointers */ unsigned yLen, ytLen, ytOffset, totalLen = 0, totalNZ = 0; /* Transform if enough samples, else just return output size. NOTE: the D4 transform will be skipped for input sizes yLen0 < 4. */ if (yLen0 >= 2) { /* Figure out total number of coefficients (= maximum size of output array including boundary padding) by summing the number of detail coeffs returned per dimension by each xform iteration. Also sum number of nonzero detail coefficients. */ for (yLen = yLen0, totalLen = 0; yLen > 3; totalLen += yLen = padD4Step2(NULL, NULL, yLen, 0), totalNZ += yLen * yLen ); /* Finally add number of approx coeffs at coarsest resolution */ totalLen += yLen; totalNZ = 3 * totalNZ + yLen * yLen; if (y && yt) { /* Apply Daubechies D4 transform, accounting for boundary padding */ for (yLen = yLen0, ytOffset = totalLen; yLen > 3; yLen = ytLen ) { /* Get num of coeff pairs returned by the next xform iteration, and calculate the offset of detail coeffs in output array */ ytOffset -= ytLen = padD4Step2(NULL, NULL, yLen, 0); /* Apply horizontal & vertical Daubechies D4 transform, swapping input and transposed output array. The resulting detail coefficients are prepended to those from the previous iteration from position ytOffset; consequently, the returned number of coefficient pairs is the input size for next iteration */ if (!padD4Step2(y, yt, yLen, ytOffset) || !padD4Step2(yt, y, yLen, ytOffset) ) return 0; } #ifdef WAVELET_FINAL_HAAR /* Apply horizontal & vertical Haar transform in penultimate and final iterations (since yLen is odd) to decompose to a single approximation coefficient y [0] [0] at the coarsest resolution; all other coeffs are details. Obviously no boundary padding is necessary here. */ for (; yLen >= 2; yLen--) { if (!padHaarStep2(y, yt, yLen) || !padHaarStep2(yt, y, yLen)) return 0; } #endif } } /* Optionally return number of nonzero coeffs */ if (numNonZero) *numNonZero = totalNZ; /* All coeffs now in y; return output size */ return totalLen; } unsigned padWaveletInvXform2 (WaveletMatrix2 y, WaveletMatrix2 yt, unsigned yLen0, unsigned yLenOrg ) /* Perform full 2D multiresolution inverse wavelet transform on array y containing (yLen0 x yLen0) boundary padded wavelet coefficients as 3-tuples. The original number of samples (yLenOrg x yLenOrg) is required for the reconstruction. An additional array yt of identical dimension to y is required as buffer for intermediate results. The reconstructed samples are returned in y. This is smaller than (yLen0 x yLen0) since the boundary coefficients from the forward transform are collapsed. The returned value is the reconstructed number of samples per dimension (yLenOrg), or 0 if the inverse transform failed. */ { unsigned yLen = 0, nextyLen, ytLen, ytOffset, i, j, d4Iter; if (yLenOrg < 2) /* Too few coefficients for reconstruction */ return 0; /* Figure out num Daubechies iterations based on original num coeffs */ for (d4Iter = 0, yLen = yLenOrg; yLen > 3; d4Iter++, yLen = padD4Step2(NULL, NULL, yLen, 0) ); #ifdef WAVELET_FINAL_HAAR /* Invert horizontal & vertical Haar transform at two coarsest levels */ for (yLen = 2; yLen <= min(3, yLen0); yLen++) if (!padHaarInvStep2(y, yt, yLen) || !padHaarInvStep2(yt, y, yLen)) return 0; #endif /* Invert horiz & vert Daubechies D4 transform on subsequent levels */ for (i = d4Iter; i; i--) { /* Get num detail coeffs and their offset for current iteration */ for (j = 0, ytOffset = yLen0, yLen = nextyLen = yLenOrg; j < i; j++) { nextyLen = yLen; ytOffset -= yLen = padD4Step2(NULL, NULL, yLen, 0); } if (!(ytLen = padD4InvStep2(y, yt, yLen, ytOffset)) || !(ytLen = padD4InvStep2(yt, y, yLen, ytOffset)) ) return 0; /* XXX: Drop excess reconstructed approximation coeffs to match number of detail coeffs in next iteration. */ yLen = min(ytLen, nextyLen); } return yLenOrg; } #ifdef WAVELET_TEST_2DPAD #include #ifdef WAVELET_TEST_mRGBE #include "mrgbe.h" #endif #define WAVELET_TEST_INIT 6 static void avgDetailCoeffs (const WaveletMatrix2 y, unsigned yLen, WaveletCoeff3 avg ) { unsigned i, j, k, numCoeffs = 0; avg [0] = avg [1] = avg [2] = 0; for (i = 0; i < yLen; i++) for (j = 0; j < yLen; j++) /* Skip zero and cleared coeffs */ if ((i > 2 || j > 2) && !coeffIsZero(y [i][j]) && !coeffIsEmpty(y [i][j]) ) { for (k = 0; k < 3; k++) avg [k] += y [i][j][k]; numCoeffs++; } for (k = 0; k < 3; k++) avg [k] /= numCoeffs; } int main (int argc, char *argv []) { int i, j, k; unsigned y0Len, yLen, numThresh = 0, numNonzero = 0; WaveletMatrix2 y0 = NULL, y = NULL, yt = NULL; FILE *dataFile = NULL; WAVELET_COEFF inData, thresh = 0; WaveletCoeff3 avgDetail; #ifdef WAVELET_TEST_mRGBE #define HUGE 1e10 mRGBE mrgbeCoeff; mRGBERange mrgbeRange; WaveletMatrix2 ymrgbe = NULL; #endif if (argc < 2) { fprintf(stderr, "%s [threshold] [dataFile]\n", argv [0]); fputs("Missing array dimension dim >= 2, " "compression threshold >= 0\n", stderr ); return -1; } if (!(y0Len = atoi(argv [1])) || y0Len < 2) { fprintf(stderr, "Invalid array dimension %d\n", y0Len); return -1; } if (argc > 2 && (thresh = atof(argv [2])) < 0) { fprintf(stderr, "Invalid threshold %.3f\n", thresh); return -1; } /* Get size of (padded) output array */ if (!(yLen = padWaveletXform2(NULL, NULL, y0Len, &numNonzero))) { fputs("Can't determine number of coefficients " "(input too short?)\n", stderr ); return -1; } /* Allocate & clear arrays for original and reconstruction */ if (!(y0 = allocWaveletMatrix2(y0Len)) || !(y = allocWaveletMatrix2(yLen)) || !(yt = allocWaveletMatrix2(yLen)) #ifdef WAVELET_TEST_mRGBE || !(ymrgbe = allocWaveletMatrix2(yLen)) #endif ) { fprintf(stderr, "Failed allocating %dx%d array\n", yLen, yLen); return -1; } clearCoeffs2(y0, y0Len); clearCoeffs2(y, yLen); clearCoeffs2(yt, yLen); if (argc > 3) { /* Load data from file; length must not exceed allocated */ if (!(dataFile = fopen(argv [3], "r"))) { fprintf(stderr, "Failed opening data file %s\n", argv [3]); return -1; } for (i = 0; i < y0Len; i++) { for (j = 0; j < y0Len; j++) { if (feof(dataFile)) { fprintf(stderr, "Premature end of file reading data from %s\n", argv [2] ); fclose(dataFile); return -1; } /* Read next float, skipping any leading whitespace */ if (fscanf(dataFile, " %lf", &inData)) { y0 [i][j][0] = y0 [i][j][1] = y0 [i][j][2] = y [i][j][0] = y [i][j][1] = y [i][j][2] = inData; } else { fprintf(stderr, "Error reading from data file %s\n", argv [2] ); fclose(dataFile); return -1; } } } fclose(dataFile); } else { /* Init input */ srand48(111); for (i = 0; i < y0Len; i++) { for (j = 0; j < y0Len; j++) { for (k = 0; k < 3; k++) { y0 [i][j][k] = y [i][j][k] = #if WAVELET_TEST_INIT == 0 /* Random data, channel-independent */ drand48(); #elif WAVELET_TEST_INIT == 1 /* Random data, indentical for all channels */ k ? y [i][j][k - 1] : drand48(); #elif WAVELET_TEST_INIT == 2 /* Monotonically increasing along axis 0 */ i; #elif WAVELET_TEST_INIT == 3 /* Monotonically increasing along axis 1 */ j; #elif WAVELET_TEST_INIT == 4 /* Monotonically increasing along both axes */ i * j; #elif WAVELET_TEST_INIT == 5 /* Monotonically increasing by linear index */ i * y0Len + j; #elif WAVELET_TEST_INIT == 6 /* Symmetric "bump" function */ (1. - fabs(j - y0Len/2. + 0.5) / (y0Len/2. - 0.5)) * (1. - fabs(i - y0Len/2. + 0.5) / (y0Len/2. - 0.5)); #endif } } } } #ifdef WAVELET_DBG puts("----------------------- FWD XFORM -------------------------\n"); #endif /* Forward xform */ if (!padWaveletXform2(y, yt, y0Len, NULL)) { fputs("Forward xform failed\n", stderr); return -1; } /* Threshold coefficients; we use hard thresholding as it's easier to * implement than soft thresholding, which requires sorting the * coefficients. NOTE: the coarsest approximation coefficient at * y [0][0] is omitted as it is essential for reconstruction! */ for (i = numThresh = 0; i < yLen; i++) for (j = 0; j < yLen; j++) { if (coeffThresh(y, i, j, thresh)) { y [i][j][0] = y [i][j][1] = y [i][j][2] = 0; numThresh++; #if 0 /* Replace thresholded values with random noise in range [-threshold, threshold] */ y [i][j][0] = y [i][j][1] = y [i][j][2] = thresh * ( 2 * drand48() - 1 ); #endif } } #ifdef WAVELET_DBG /* Dump coefficients */ puts("----------------------- COEFFICIENTS -------------------------\n"); dumpCoeffs2(y, NULL, yLen, 0, thresh); printf("\n%d/%d nonzero coefficients", numNonzero, yLen * yLen); avgDetailCoeffs(y, yLen, avgDetail); printf("\nAvg = [%.3g, %.3g, %.3g]", avgDetail [0], avgDetail [1], avgDetail [2] ); if (numThresh) printf("\n%d/%d coefficients thresholded = %.1f%% compression", numThresh, yLen * yLen, 100. * numThresh / (yLen * yLen) ); #endif #ifdef WAVELET_TEST_mRGBE /* Test mRGBE coefficient encoding */ mrgbeRange.min [0] = mrgbeRange.min [1] = mrgbeRange.min [2] = HUGE; mrgbeRange.max [0] = mrgbeRange.max [1] = mrgbeRange.max [2] = 0; clearCoeffs2(ymrgbe, yLen); /* Find min/max coeff and init mRGBE range */ for (i = 0; i < yLen; i++) for (j = 0; j < yLen; j++) if (!coeffIsEmpty(y [i][j]) && !coeffIsZero(y [i][j])) for (k = 0; k < 3; k++) { /* Skip empty (NaN) coeffs */ inData = fabs(y [i][j][k]); if (inData < mrgbeRange.min [k]) mrgbeRange.min [k] = inData; if (inData > mrgbeRange.max [k]) mrgbeRange.max [k] = inData; } mRGBEinit(&mrgbeRange, mrgbeRange.min, mrgbeRange.max); /* Encode coeffs to mRGBE and back, but preserve approximation * coefficients */ for (i = 0; i < yLen; i++) for (j = 0; j < yLen; j++) if (!coeffIsEmpty(y [i][j]) && !coeffIsZero(y [i][j])) { mrgbeCoeff = mRGBEencode(y [i][j], &mrgbeRange, 0); mRGBEdecode(mrgbeCoeff, &mrgbeRange, ymrgbe [i][j]); } else for (k = 0; k < 3; k++) ymrgbe [i][j][k] = y [i][j][k]; #ifdef WAVELET_DBG /* Dump mRGBE-decoded coefficients */ puts("\n\n-------------------- mRGBE COEFFICIENTS ----------------------\n"); dumpCoeffs2(ymrgbe, NULL, yLen, yLen, 0); avgDetailCoeffs(y, yLen, avgDetail); printf("\nAvg = [%.3g, %.3g, %.3g]", avgDetail [0], avgDetail [1], avgDetail [2] ); #endif #endif #ifdef WAVELET_DBG puts("\n----------------------- INV XFORM -------------------------\n"); #endif /* Inverse xform (also using mRGBE coeffs if enabled) */ if (!padWaveletInvXform2(y, yt, yLen, y0Len) #ifdef WAVELET_TEST_mRGBE || !padWaveletInvXform2(ymrgbe, yt, yLen, y0Len) #endif ) { fputs("\nInverse xform failed\n", stderr); return -1; } #ifdef WAVELET_DBG puts("\n--------------------- ORIG vs. INV XFORM ------------------------\n"); dumpCoeffs2(y0, y, y0Len, y0Len, 0); #endif printf("\nAvg RMSE = %.2f\n", rmseCoeffs2(y0, y, y0Len)); #ifdef WAVELET_TEST_mRGBE #ifdef WAVELET_DBG puts("\n------------------ ORIG vs. INV XFORM + mRGBE -------------------\n"); dumpCoeffs2(y0, ymrgbe, y0Len, y0Len, 0); #endif printf("\nAvg RMSE with mRGBE enc = %.2f\n", rmseCoeffs2(y0, ymrgbe, y0Len) ); #endif freeWaveletMatrix2(y0, y0Len); freeWaveletMatrix2(y, yLen); freeWaveletMatrix2(yt, yLen); #ifdef WAVELET_TEST_mRGBE freeWaveletMatrix2(ymrgbe, yLen); #endif return 0; } #endif /* WAVELET_TEST_2DPAD */