/* * integration.c * * Integration of intensities * * Copyright © 2012-2013 Deutsches Elektronen-Synchrotron DESY, * a research centre of the Helmholtz Association. * * Authors: * 2010-2013 Thomas White * * This file is part of CrystFEL. * * CrystFEL is free software: you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation, either version 3 of the License, or * (at your option) any later version. * * CrystFEL is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with CrystFEL. If not, see . * */ #ifdef HAVE_CONFIG_H #include #endif #include #include #include #include #include #include #include "reflist.h" #include "cell.h" #include "crystal.h" #include "cell-utils.h" #include "geometry.h" #include "image.h" #include "peaks.h" #include "integration.h" static void check_eigen(gsl_vector *e_val) { int i; double vmax, vmin; const int n = e_val->size; const double max_condition = 1e6; const int verbose = 0; if ( verbose ) STATUS("Eigenvalues:\n"); vmin = +INFINITY; vmax = 0.0; for ( i=0; i vmax ) vmax = val; if ( val < vmin ) vmin = val; } for ( i=0; i vmax ) vmax = val; if ( val < vmin ) vmin = val; } if ( verbose ) { STATUS("Condition number: %e / %e = %5.2f\n", vmax, vmin, vmax/vmin); } } static gsl_vector *solve_svd(gsl_vector *v, gsl_matrix *M) { gsl_matrix *s_vec; gsl_vector *s_val; int err, n; gsl_vector *shifts; n = v->size; if ( v->size != M->size1 ) return NULL; if ( v->size != M->size2 ) return NULL; s_val = gsl_vector_calloc(n); s_vec = gsl_matrix_calloc(n, n); err = gsl_linalg_SV_decomp_jacobi(M, s_vec, s_val); if ( err ) { ERROR("SVD failed: %s\n", gsl_strerror(err)); gsl_matrix_free(s_vec); gsl_vector_free(s_val); return NULL; } /* "M" is now "U" */ check_eigen(s_val); shifts = gsl_vector_calloc(n); err = gsl_linalg_SV_solve(M, s_vec, s_val, v, shifts); if ( err ) { ERROR("Matrix solution failed: %s\n", gsl_strerror(err)); gsl_matrix_free(s_vec); gsl_vector_free(s_val); gsl_vector_free(shifts); return NULL; } gsl_matrix_free(s_vec); gsl_vector_free(s_val); return shifts; } enum boxmask_val { BM_IG, BM_BG, BM_PK }; struct intcontext { int halfw; int w; enum boxmask_val *bm; /* Box mask */ struct image *image; gsl_matrix *bgm; /* Background estimation matrix */ struct peak_box *boxes; int n_boxes; int max_boxes; /* Peak region sums */ double pks_p2; double pks_q2; double pks_pq; double pks_p; double pks_q; int m; int n_reference_profiles; double **reference_profiles; double **reference_den; }; struct peak_box { int cfs; /* Coordinates of corner */ int css; enum boxmask_val *bm; /* Box mask */ int pn; /* Panel number */ struct panel *p; /* The panel itself */ /* Fitted background parameters */ double a; double b; double c; /* Measured intensity (tentative, profile fitted or otherwise) */ double intensity; int rp; /* Reference profile number */ Reflection *refl; int verbose; }; static void addm(gsl_matrix *m, int i, int j, double val) { double v = gsl_matrix_get(m, i, j); gsl_matrix_set(m, i, j, v+val); } static void addv(gsl_vector *v, int i, double val) { double k = gsl_vector_get(v, i); gsl_vector_set(v, i, k+val); } static float boxi(struct intcontext *ic, struct peak_box *bx, int p, int q) { int fs, ss; fs = bx->cfs + p; ss = bx->css + q; assert(fs >= 0); assert(fs < bx->p->w); assert(ss >= 0); assert(ss < bx->p->h); assert(p >= 0); assert(p < ic->w); assert(q >= 0); assert(q < ic->w); return ic->image->dp[bx->pn][fs + bx->p->w*ss]; } static void show_peak_box(struct intcontext *ic, struct peak_box *bx) { int q; printf("Pixel values "); printf("Box flags "); printf("Fitted background\n"); for ( q=ic->w-1; q>=0; q-- ) { int p; for ( p=0; pw; p++ ) { printf("%5.0f ", boxi(ic, bx, p, q)); } printf(" "); for ( p=0; pw; p++ ) { printf("%i ", bx->bm[p+q*ic->w]); } printf(" "); for ( p=0; pw; p++ ) { printf("%5.0f ", bx->a*p + bx->b*q + bx->c); } printf("\n"); } printf("Reference profile number %i\n", bx->rp); } static void show_reference_profile(struct intcontext *ic, int i) { int q; printf("Reference profile number %i:\n", i); for ( q=ic->w-1; q>=0; q-- ) { int p; for ( p=0; pw; p++ ) { printf("%3.0f ", ic->reference_profiles[i][p+ic->w*q]); } printf("\n"); } } static void fit_bg(struct intcontext *ic, struct peak_box *bx) { int p, q; gsl_vector *v; gsl_vector *ans; gsl_matrix *M; v = gsl_vector_calloc(3); for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { double bi; if ( bx->bm[p + ic->w*q] == BM_BG ) { bi = boxi(ic, bx, p, q); addv(v, 0, bi*p); addv(v, 1, bi*q); addv(v, 2, bi); } } } if ( bx->verbose ) { show_matrix_eqn(ic->bgm, v, 3); } M = gsl_matrix_alloc(3, 3); gsl_matrix_memcpy(M, ic->bgm); ans = solve_svd(v, M); gsl_vector_free(v); gsl_matrix_free(M); bx->a = gsl_vector_get(ans, 0); bx->b = gsl_vector_get(ans, 1); bx->c = gsl_vector_get(ans, 2); gsl_vector_free(ans); if ( bx->verbose ) { show_peak_box(ic, bx); } } static void zero_profiles(struct intcontext *ic) { int i; for ( i=0; in_reference_profiles; i++ ) { int p, q; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { ic->reference_profiles[i][p+ic->w*q] = 0.0; } } } } static int alloc_boxes(struct intcontext *ic, int new_max_boxes) { struct peak_box *boxes_new; boxes_new = realloc(ic->boxes, sizeof(struct peak_box)*new_max_boxes); if ( boxes_new == NULL ) return 1; ic->boxes = boxes_new; ic->max_boxes = new_max_boxes; return 0; } static int init_intcontext(struct intcontext *ic) { int p, q; int i; ic->w = 2*ic->halfw + 1; ic->bm = malloc(ic->w * ic->w * sizeof(enum boxmask_val)); if ( ic->bm == NULL ) { ERROR("Failed to allocate box mask.\n"); return 1; } ic->bgm = gsl_matrix_calloc(3, 3); if ( ic->bgm == NULL ) { ERROR("Failed to initialise matrix.\n"); return 1; } ic->pks_p2 = 0.0; ic->pks_q2 = 0.0; ic->pks_pq = 0.0; ic->pks_p = 0.0; ic->pks_q = 0.0; ic->m = 0; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { if ( (p==0) || (q==0) || (p==ic->w-1) || (q==ic->w-1) ) { ic->bm[p + ic->w*q] = BM_BG; } else { ic->bm[p + ic->w*q] = BM_PK; } switch ( ic->bm[p + ic->w*q] ) { case BM_IG : break; case BM_BG : addm(ic->bgm, 0, 0, p*p); addm(ic->bgm, 0, 1, p*q); addm(ic->bgm, 0, 2, p); addm(ic->bgm, 1, 0, p*q); addm(ic->bgm, 1, 1, q*q); addm(ic->bgm, 1, 2, q); addm(ic->bgm, 2, 0, p); addm(ic->bgm, 2, 1, q); addm(ic->bgm, 2, 2, 1); break; case BM_PK : ic->pks_p2 += p*p; ic->pks_q2 += q*q; ic->pks_pq += p*q; ic->pks_p += p; ic->pks_q += q; ic->m++; break; } } } /* How many reference profiles? */ ic->n_reference_profiles = ic->image->det->n_panels; ic->reference_profiles = calloc(ic->n_reference_profiles, sizeof(double *)); if ( ic->reference_profiles == NULL ) return 1; ic->reference_den = calloc(ic->n_reference_profiles, sizeof(double *)); if ( ic->reference_den == NULL ) return 1; for ( i=0; in_reference_profiles; i++ ) { ic->reference_profiles[i] = malloc(ic->w*ic->w*sizeof(double)); if ( ic->reference_profiles[i] == NULL ) return 1; ic->reference_den[i] = malloc(ic->w*ic->w*sizeof(double)); if ( ic->reference_den[i] == NULL ) return 1; } zero_profiles(ic); ic->boxes = NULL; ic->n_boxes = 0; ic->max_boxes = 0; if ( alloc_boxes(ic, 32) ) { return 1; } return 0; } static struct peak_box *add_box(struct intcontext *ic) { int idx; if ( ic->n_boxes == ic->max_boxes ) { if ( alloc_boxes(ic, ic->max_boxes+32) ) { return NULL; } } idx = ic->n_boxes++; ic->boxes[idx].verbose = 0; return &ic->boxes[idx]; } static void delete_box(struct intcontext *ic, struct peak_box *bx) { int i; int found = 0; for ( i=0; in_boxes; i++ ) { if ( &ic->boxes[i] == bx ) { found = 1; break; } } if ( !found ) { ERROR("Couldn't find box %p in context %p\n", bx, ic); return; } memmove(&ic->boxes[i], &ic->boxes[i+1], (ic->n_boxes-i-1)*sizeof(struct peak_box)); ic->n_boxes--; } static double tentative_intensity(struct intcontext *ic, struct peak_box *bx) { int p, q; double intensity = 0.0; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { if ( bx->bm[p + ic->w*q] != BM_PK ) continue; intensity += boxi(ic, bx, p, q); } } intensity -= bx->a * ic->pks_p; intensity -= bx->b * ic->pks_q; intensity -= bx->c * ic->m; return intensity; } static void observed_position(struct intcontext *ic, struct peak_box *bx, double *pos_p, double *pos_q) { int p, q; double num_p = 0.0; double num_q = 0.0; double den = 0.0; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { int bi; if ( bx->bm[p + ic->w*q] != BM_PK ) continue; bi = boxi(ic, bx, p, q); num_p += bi*p; num_q += bi*q; den += bi; } } num_p += -bx->a*ic->pks_p2 - bx->b*ic->pks_pq - bx->c*ic->pks_p; num_q += -bx->a*ic->pks_q2 - bx->b*ic->pks_pq - bx->c*ic->pks_q; den += -bx->a*ic->pks_p - bx->b*ic->pks_q - bx->c; *pos_p = num_p / den; *pos_q = num_q / den; } static void add_to_reference_profile(struct intcontext *ic, struct peak_box *bx) { int p, q; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { double val; float bi; if ( bx->bm[p + ic->w*q] == BM_IG ) continue; bi = boxi(ic, bx, p, q); val = bi*bx->intensity; val -= p*bx->intensity*bx->a + q*bx->intensity*bx->b + bx->intensity*bx->c; ic->reference_profiles[bx->rp][p+ic->w*q] += val; ic->reference_den[bx->rp][p+ic->w*q] += pow(bx->intensity, 2.0); } } } static void calculate_reference_profiles(struct intcontext *ic) { int i; for ( i=0; in_reference_profiles; i++ ) { int p, q; double max = 0.0; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { double den; den = ic->reference_den[i][p+ic->w*q]; ic->reference_profiles[i][p+ic->w*q] /= den; if ( ic->reference_profiles[i][p+ic->w*q] > max ) { max = ic->reference_profiles[i][p+ic->w*q]; } } } max /= 100.0; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { ic->reference_profiles[i][p+ic->w*q] /= max; } } } show_reference_profile(ic, 2); //for ( i=0; in_reference_profiles; i++ ) { // show_reference_profile(ic, i); //} } static int check_box(struct intcontext *ic, struct peak_box *bx) { int p, q; int n_pk = 0; int n_bg = 0; bx->bm = malloc(ic->w*ic->w*sizeof(int)); if ( bx->bm == NULL ) return 1; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { int fs, ss; fs = bx->cfs + p; ss = bx->css + q; assert(fs >= 0); assert(fs < bx->p->w); assert(ss >= 0); assert(ss < bx->p->h); assert(p >= 0); assert(p < ic->w); assert(q >= 0); assert(q < ic->w); bx->bm[p+ic->w*q] = ic->bm[p+ic->w*q]; if ( ic->image->bad[bx->pn][fs + bx->p->w*ss] ) { bx->bm[p+ic->w*q] = BM_IG; } if ( bx->bm[p+ic->w*q] == BM_PK ) n_pk++; if ( bx->bm[p+ic->w*q] == BM_BG ) n_bg++; } } if ( n_pk < 4 ) return 1; if ( n_bg < 4 ) return 1; return 0; } static double fit_intensity(struct intcontext *ic, struct peak_box *bx) { int p, q; double sum = 0.0; double sp = 0.0; double den = 0.0; for ( p=0; pw; p++ ) { for ( q=0; qw; q++ ) { double bi, P; if ( bx->bm[p + ic->w*q] != BM_PK ) continue; bi = boxi(ic, bx, p, q); P = ic->reference_profiles[bx->rp][p+ic->w*q]; sum += bi*P; sum += - bx->a*p*P - bx->b*q*P - bx->c*P; sp += P; den += pow(P, 2.0); } } return sum*sp / den; } static void measure_all_intensities(RefList *list, struct image *image) { Reflection *refl; RefListIterator *iter; struct intcontext ic; int i; ic.halfw = 4; ic.image = image; if ( init_intcontext(&ic) ) { ERROR("Failed to initialise integration.\n"); return; } for ( refl = first_refl(list, &iter); refl != NULL; refl = next_refl(refl, iter) ) { double pfs, pss; double pos_p, pos_q; signed int h, k, l; struct peak_box *bx; int pn; struct panel *p; int fid_fs, fid_ss; /* Center coordinates, rounded, * in overall data block */ int cfs, css; /* Corner coordinates */ set_redundancy(refl, 0); get_detector_pos(refl, &pfs, &pss); fid_fs = lrint(pfs); fid_ss = lrint(pss); pn = find_panel_number(image->det, fid_fs, fid_ss); p = &image->det->panels[pn]; cfs = (fid_fs-p->min_fs) - ic.halfw; css = (fid_ss-p->min_ss) - ic.halfw; if ( (cfs + ic.w >= p->w) || (css + ic.w >= p->h ) ) { continue; } if ( (cfs < 0) || (css < 0 ) ) continue; bx = add_box(&ic); bx->refl = refl; bx->cfs = cfs; bx->css = css; bx->p = p; bx->pn = pn; if ( check_box(&ic, bx) ) { delete_box(&ic, bx); continue; } get_indices(refl, &h, &k, &l); if ( (h==-24) && (k==6) && (l==-12) ) { bx->verbose = 1; } fit_bg(&ic, bx); observed_position(&ic, bx, &pos_p, &pos_q); pos_p -= ic.halfw; pos_q -= ic.halfw; if ( bx->verbose ) { STATUS("Position error %f %f\n", pos_p, pos_q); } bx->intensity = tentative_intensity(&ic, bx); set_intensity(refl, bx->intensity); /* Which reference profile? */ bx->rp = bx->pn; add_to_reference_profile(&ic, bx); } calculate_reference_profiles(&ic); for ( i=0; iverbose ) { STATUS("%f -> ", bx->intensity); } bx->intensity = fit_intensity(&ic, bx); if ( isnan(bx->intensity) ) { signed int h, k, l; get_indices(bx->refl, &h, &k, &l); STATUS("%i %i %i\n", h, k, l); STATUS("panel %s\n", image->det->panels[bx->pn].name); show_peak_box(&ic, bx); } set_intensity(bx->refl, bx->intensity); set_redundancy(bx->refl, 1); if ( bx->verbose ) { STATUS("%f\n", bx->intensity); } } } static void estimate_mosaicity(Crystal *cr, struct image *image) { int msteps = 50; int i; const double mest = crystal_get_mosaicity(cr); const double mmax = 2.0 * mest; RefList *list; STATUS("Initial estimate: m = %f\n", mest); crystal_set_mosaicity(cr, mmax); list = find_intersections(image, cr); crystal_set_reflections(cr, list); measure_all_intensities(list, image); for ( i=1; i<=msteps; i++ ) { /* "m" varies from just over zero up to 2x the given estimate */ Reflection *refl; RefListIterator *iter; const double m = mmax*((double)i/msteps); int n_gained = 0; int n_lost = 0; double i_gained = 0.0; double i_lost = 0.0; crystal_set_mosaicity(cr, m); update_partialities(cr, PMODEL_SPHERE); for ( refl = first_refl(list, &iter); refl != NULL; refl = next_refl(refl, iter) ) { if ( get_redundancy(refl) == 0 ) { if ( get_temp1(refl) > 0.0 ) { i_lost += get_intensity(refl); n_lost++; } set_temp1(refl, -1.0); } else if ( get_temp1(refl) < 0.0 ) { i_gained += get_intensity(refl); n_gained++; set_temp1(refl, 1.0); } } if ( i > 1 ) { STATUS("%.2e %10.2f %4i %10.2f %4i %10.2f\n", m, i_gained, n_gained, i_lost, n_lost, i_gained - i_lost); } } } struct integr_ind { double res; Reflection *refl; }; static int compare_resolution(const void *av, const void *bv) { const struct integr_ind *a = av; const struct integr_ind *b = bv; return a->res > b->res; } static struct integr_ind *sort_reflections(RefList *list, UnitCell *cell, int *np) { struct integr_ind *il; Reflection *refl; RefListIterator *iter; int i, n; *np = 0; /* For now */ n = num_reflections(list); if ( n == 0 ) return NULL; il = calloc(n, sizeof(struct integr_ind)); if ( il == NULL ) return NULL; i = 0; for ( refl = first_refl(list, &iter); refl != NULL; refl = next_refl(refl, iter) ) { signed int h, k, l; double res; if ( get_redundancy(refl) == 0 ) continue; get_indices(refl, &h, &k, &l); res = resolution(cell, h, k, l); il[i].res = res; il[i].refl = refl; i++; } qsort(il, i, sizeof(struct integr_ind), compare_resolution); *np = i; return il; } static void estimate_resolution(RefList *reflections, Crystal *cr, struct image *image) { struct integr_ind *il; int n, i; int score = 1000; /* FIXME */ int cutoff = 0; double limit = 0.0; if ( num_reflections(reflections) == 0 ) return; il = sort_reflections(reflections, crystal_get_cell(cr), &n); if ( il == NULL ) { ERROR("Couldn't sort reflections\n"); return; } for ( i=0; i 3.0 ) { score++; } else { score--; } //STATUS("%5.2f A, %5.2f, %i\n", 1e10/il[i].res, snr, score); if ( score == 0 ) { limit = il[i].res; cutoff = 1; } } crystal_set_resolution_limit(cr, limit); free(il); } static void integrate_refine(Crystal *cr, struct image *image, int use_closer, double min_snr, double ir_inn, double ir_mid, double ir_out, int integrate_saturated, int **bgMasks) { RefList *reflections; /* Create initial list of reflections with nominal parameters */ reflections = find_intersections(image, cr); measure_all_intensities(reflections, image); /* Find resolution limit of pattern using this list */ estimate_resolution(reflections, cr, image); reflist_free(reflections); STATUS("Initial resolution estimate = %.2f nm^-1 or %.2f A\n", crystal_get_resolution_limit(cr)/1e9, 1e9 / crystal_get_resolution_limit(cr)); /* Estimate the mosaicity of the crystal using this resolution limit */ estimate_mosaicity(cr, image); /* Create new list of reflections with refined mosaicity */ reflections = find_intersections(image, cr); measure_all_intensities(reflections, image); estimate_resolution(reflections, cr, image); } static void integrate_rings(Crystal *cr, struct image *image, int use_closer, double min_snr, double ir_inn, double ir_mid, double ir_out, int integrate_saturated, int **bgMasks) { RefList *list; Reflection *refl; RefListIterator *iter; UnitCell *cell; int n_saturated = 0; double limit = 0.0; list = find_intersections(image, cr); if ( list == NULL ) return; if ( num_reflections(list) == 0 ) return; cell = crystal_get_cell(cr); for ( refl = first_refl(list, &iter); refl != NULL; refl = next_refl(refl, iter) ) { double fs, ss, intensity; double d; int idx; double sigma, snr; double pfs, pss; int r; signed int h, k, l; struct panel *p; int pnum, j, found; int saturated; double one_over_d; get_detector_pos(refl, &pfs, &pss); get_indices(refl, &h, &k, &l); /* Is there a really close feature which was detected? */ if ( use_closer ) { struct imagefeature *f; if ( image->features != NULL ) { f = image_feature_closest(image->features, pfs, pss, &d, &idx); } else { f = NULL; } /* FIXME: Horrible hardcoded value */ if ( (f != NULL) && (d < 10.0) ) { double exe; exe = get_excitation_error(refl); pfs = f->fs; pss = f->ss; set_detector_pos(refl, exe, pfs, pss); } } p = find_panel(image->det, pfs, pss); if ( p == NULL ) continue; /* Next peak */ found = 0; for ( j=0; jdet->n_panels; j++ ) { if ( &image->det->panels[j] == p ) { pnum = j; found = 1; break; } } if ( !found ) { ERROR("Couldn't find panel %p in list.\n", p); return; } r = integrate_peak(image, pfs, pss, &fs, &ss, &intensity, &sigma, ir_inn, ir_mid, ir_out, bgMasks[pnum], &saturated); if ( !r && saturated ) { n_saturated++; if ( !integrate_saturated ) r = 1; } /* I/sigma(I) cutoff * Rejects reflections below --min-integration-snr, or if the * SNR is clearly silly. Silly indicates that the intensity * was zero. */ snr = fabs(intensity)/sigma; if ( !r && (isnan(snr) || (snr < min_snr)) ) { r = 1; } /* Record intensity and set redundancy to 1 on success */ if ( !r ) { set_intensity(refl, intensity); set_esd_intensity(refl, sigma); set_redundancy(refl, 1); } else { set_redundancy(refl, 0); } one_over_d = resolution(cell, h, k, l); if ( one_over_d > limit ) limit = one_over_d; } crystal_set_num_saturated_reflections(cr, n_saturated); crystal_set_resolution_limit(cr, limit); crystal_set_reflections(cr, list); } void integrate_all(struct image *image, IntegrationMethod meth, int use_closer, double min_snr, double ir_inn, double ir_mid, double ir_out, int integrate_saturated) { int i; int **bgMasks; /* Make background masks for all panels */ bgMasks = calloc(image->det->n_panels, sizeof(int *)); if ( bgMasks == NULL ) { ERROR("Couldn't create list of background masks.\n"); return; } for ( i=0; idet->n_panels; i++ ) { int *mask; mask = make_BgMask(image, &image->det->panels[i], ir_inn); if ( mask == NULL ) { ERROR("Couldn't create background mask.\n"); return; } bgMasks[i] = mask; } for ( i=0; in_crystals; i++ ) { switch ( meth & INTEGRATION_METHOD_MASK ) { case INTEGRATION_NONE : return; case INTEGRATION_RINGS : integrate_rings(image->crystals[i], image, use_closer, min_snr, ir_inn, ir_mid, ir_out, integrate_saturated, bgMasks); return; case INTEGRATION_REFINE : integrate_refine(image->crystals[i], image, use_closer, min_snr, ir_inn, ir_mid, ir_out, integrate_saturated, bgMasks); return; default : ERROR("Unrecognised integration method %i\n", meth); return; } } for ( i=0; idet->n_panels; i++ ) { free(bgMasks[i]); } free(bgMasks); } IntegrationMethod integration_method(const char *str, int *err) { int n, i; char **methods; IntegrationMethod meth = INTEGRATION_NONE; if ( err != NULL ) *err = 0; n = assplode(str, ",-", &methods, ASSPLODE_NONE); for ( i=0; i