aom_dsp/flow_estimation/ransac.c - aom - Git at Google

 /*
  * Copyright (c) 2016, Alliance for Open Media. All rights reserved.
  *
  * This source code is subject to the terms of the BSD 2 Clause License and
  * the Alliance for Open Media Patent License 1.0. If the BSD 2 Clause License
  * was not distributed with this source code in the LICENSE file, you can
  * obtain it at www.aomedia.org/license/software. If the Alliance for Open
  * Media Patent License 1.0 was not distributed with this source code in the
  * PATENTS file, you can obtain it at www.aomedia.org/license/patent.
  */

 #include <memory.h>
 #include <math.h>
 #include <time.h>
 #include <stdio.h>
 #include <stdbool.h>
 #include <string.h>
 #include <assert.h>

 #include "aom_dsp/flow_estimation/ransac.h"
 #include "aom_dsp/mathutils.h"
 #include "aom_mem/aom_mem.h"

 // TODO(rachelbarker): Remove dependence on code in av1/encoder/
 #include "av1/encoder/random.h"

 #define MAX_MINPTS 4
 #define MINPTS_MULTIPLIER 5

 #define INLIER_THRESHOLD 1.25
 #define INLIER_THRESHOLD_SQUARED (INLIER_THRESHOLD * INLIER_THRESHOLD)

 // Number of initial models to generate
 #define NUM_TRIALS 20

 // Number of times to refine the best model found
 #define NUM_REFINES 5

 // Flag to enable functions for finding TRANSLATION type models.
 //
 // These modes are not considered currently due to a spec bug (see comments
 // in gm_get_motion_vector() in av1/common/mv.h). Thus we don't need to compile
 // the corresponding search functions, but it is nice to keep the source around
 // but disabled, for completeness.
 #define ALLOW_TRANSLATION_MODELS 0

 typedef struct {
   int num_inliers;
   double sse;  // Sum of squared errors of inliers
   int *inlier_indices;
 } RANSAC_MOTION;

 ////////////////////////////////////////////////////////////////////////////////
 // ransac
 typedef bool (*FindTransformationFunc)(const Correspondence *points,
                                        const int *indices, int num_indices,
                                        double *params);
 typedef void (*ScoreModelFunc)(const double *mat, const Correspondence *points,
                                int num_points, RANSAC_MOTION *model);

 // vtable-like structure which stores all of the information needed by RANSAC
 // for a particular model type
 typedef struct {
   FindTransformationFunc find_transformation;
   ScoreModelFunc score_model;

   // The minimum number of points which can be passed to find_transformation
   // to generate a model.
   //
   // This should be set as small as possible. This is due to an observation
   // from section 4 of "Optimal Ransac" by A. Hast, J. Nysjö and
   // A. Marchetti (https://dspace5.zcu.cz/bitstream/11025/6869/1/Hast.pdf):
   // using the minimum possible number of points in the initial model maximizes
   // the chance that all of the selected points are inliers.
   //
   // That paper proposes a method which can deal with models which are
   // contaminated by outliers, which helps in cases where the inlier fraction
   // is low. However, for our purposes, global motion only gives significant
   // gains when the inlier fraction is high.
   //
   // So we do not use the method from this paper, but we do find that
   // minimizing the number of points used for initial model fitting helps
   // make the best use of the limited number of models we consider.
   int minpts;
 } RansacModelInfo;

 #if ALLOW_TRANSLATION_MODELS
 static void score_translation(const double *mat, const Correspondence *points,
                               int num_points, RANSAC_MOTION *model) {
   model->num_inliers = 0;
   model->sse = 0.0;

   for (int i = 0; i < num_points; ++i) {
     const double x1 = points[i].x;
     const double y1 = points[i].y;
     const double x2 = points[i].rx;
     const double y2 = points[i].ry;

     const double proj_x = x1 + mat[0];
     const double proj_y = y1 + mat[1];

     const double dx = proj_x - x2;
     const double dy = proj_y - y2;
     const double sse = dx * dx + dy * dy;

     if (sse < INLIER_THRESHOLD_SQUARED) {
       model->inlier_indices[model->num_inliers++] = i;
       model->sse += sse;
     }
   }
 }
 #endif  // ALLOW_TRANSLATION_MODELS

 static void score_affine(const double *mat, const Correspondence *points,
                          int num_points, RANSAC_MOTION *model) {
   model->num_inliers = 0;
   model->sse = 0.0;

   for (int i = 0; i < num_points; ++i) {
     const double x1 = points[i].x;
     const double y1 = points[i].y;
     const double x2 = points[i].rx;
     const double y2 = points[i].ry;

     const double proj_x = mat[2] * x1 + mat[3] * y1 + mat[0];
     const double proj_y = mat[4] * x1 + mat[5] * y1 + mat[1];

     const double dx = proj_x - x2;
     const double dy = proj_y - y2;
     const double sse = dx * dx + dy * dy;

     if (sse < INLIER_THRESHOLD_SQUARED) {
       model->inlier_indices[model->num_inliers++] = i;
       model->sse += sse;
     }
   }
 }

 #if ALLOW_TRANSLATION_MODELS
 static bool find_translation(const Correspondence *points, const int *indices,
                              int num_indices, double *params) {
   double sumx = 0;
   double sumy = 0;

   for (int i = 0; i < num_indices; ++i) {
     int index = indices[i];
     const double sx = points[index].x;
     const double sy = points[index].y;
     const double dx = points[index].rx;
     const double dy = points[index].ry;

     sumx += dx - sx;
     sumy += dy - sy;
   }

   params[0] = sumx / np;
   params[1] = sumy / np;
   params[2] = 1;
   params[3] = 0;
   params[4] = 0;
   params[5] = 1;
   return true;
 }
 #endif  // ALLOW_TRANSLATION_MODELS

 static bool find_rotzoom(const Correspondence *points, const int *indices,
                          int num_indices, double *params) {
   const int n = 4;    // Size of least-squares problem
   double mat[4 * 4];  // Accumulator for A'A
   double y[4];        // Accumulator for A'b
   double a[4];        // Single row of A
   double b;           // Single element of b

   least_squares_init(mat, y, n);
   for (int i = 0; i < num_indices; ++i) {
     int index = indices[i];
     const double sx = points[index].x;
     const double sy = points[index].y;
     const double dx = points[index].rx;
     const double dy = points[index].ry;

     a[0] = 1;
     a[1] = 0;
     a[2] = sx;
     a[3] = sy;
     b = dx;
     least_squares_accumulate(mat, y, a, b, n);

     a[0] = 0;
     a[1] = 1;
     a[2] = sy;
     a[3] = -sx;
     b = dy;
     least_squares_accumulate(mat, y, a, b, n);
   }

   // Fill in params[0] .. params[3] with output model
   if (!least_squares_solve(mat, y, params, n)) {
     return false;
   }

   // Fill in remaining parameters
   params[4] = -params[3];
   params[5] = params[2];

   return true;
 }

 static bool find_affine(const Correspondence *points, const int *indices,
                         int num_indices, double *params) {
   // Note: The least squares problem for affine models is 6-dimensional,
   // but it splits into two independent 3-dimensional subproblems.
   // Solving these two subproblems separately and recombining at the end
   // results in less total computation than solving the 6-dimensional
   // problem directly.
   //
   // The two subproblems correspond to all the parameters which contribute
   // to the x output of the model, and all the parameters which contribute
   // to the y output, respectively.

   const int n = 3;       // Size of each least-squares problem
   double mat[2][3 * 3];  // Accumulator for A'A
   double y[2][3];        // Accumulator for A'b
   double x[2][3];        // Output vector
   double a[2][3];        // Single row of A
   double b[2];           // Single element of b

   least_squares_init(mat[0], y[0], n);
   least_squares_init(mat[1], y[1], n);
   for (int i = 0; i < num_indices; ++i) {
     int index = indices[i];
     const double sx = points[index].x;
     const double sy = points[index].y;
     const double dx = points[index].rx;
     const double dy = points[index].ry;

     a[0][0] = 1;
     a[0][1] = sx;
     a[0][2] = sy;
     b[0] = dx;
     least_squares_accumulate(mat[0], y[0], a[0], b[0], n);

     a[1][0] = 1;
     a[1][1] = sx;
     a[1][2] = sy;
     b[1] = dy;
     least_squares_accumulate(mat[1], y[1], a[1], b[1], n);
   }

   if (!least_squares_solve(mat[0], y[0], x[0], n)) {
     return false;
   }
   if (!least_squares_solve(mat[1], y[1], x[1], n)) {
     return false;
   }

   // Rearrange least squares result to form output model
   params[0] = x[0][0];
   params[1] = x[1][0];
   params[2] = x[0][1];
   params[3] = x[0][2];
   params[4] = x[1][1];
   params[5] = x[1][2];

   return true;
 }

 // Return -1 if 'a' is a better motion, 1 if 'b' is better, 0 otherwise.
 static int compare_motions(const void *arg_a, const void *arg_b) {
   const RANSAC_MOTION *motion_a = (RANSAC_MOTION *)arg_a;
   const RANSAC_MOTION *motion_b = (RANSAC_MOTION *)arg_b;

   if (motion_a->num_inliers > motion_b->num_inliers) return -1;
   if (motion_a->num_inliers < motion_b->num_inliers) return 1;
   if (motion_a->sse < motion_b->sse) return -1;
   if (motion_a->sse > motion_b->sse) return 1;
   return 0;
 }

 static bool is_better_motion(const RANSAC_MOTION *motion_a,
                              const RANSAC_MOTION *motion_b) {
   return compare_motions(motion_a, motion_b) < 0;
 }

 // Returns true on success, false on error
 static bool ransac_internal(const Correspondence *matched_points, int npoints,
                             MotionModel *motion_models, int num_desired_motions,
                             const RansacModelInfo *model_info,
                             bool *mem_alloc_failed) {
   assert(npoints >= 0);
   int i = 0;
   int minpts = model_info->minpts;
   bool ret_val = true;

   unsigned int seed = (unsigned int)npoints;

   int indices[MAX_MINPTS] = { 0 };

   // Store information for the num_desired_motions best transformations found
   // and the worst motion among them, as well as the motion currently under
   // consideration.
   RANSAC_MOTION *motions, *worst_kept_motion = NULL;
   RANSAC_MOTION current_motion;

   // Store the parameters and the indices of the inlier points for the motion
   // currently under consideration.
   double params_this_motion[MAX_PARAMDIM];

   // Initialize output models, as a fallback in case we can't find a model
   for (i = 0; i < num_desired_motions; i++) {
     memcpy(motion_models[i].params, kIdentityParams,
            MAX_PARAMDIM * sizeof(*(motion_models[i].params)));
     motion_models[i].num_inliers = 0;
   }

   if (npoints < minpts * MINPTS_MULTIPLIER || npoints == 0) {
     return false;
   }

   int min_inliers = AOMMAX((int)(MIN_INLIER_PROB * npoints), minpts);

   motions =
       (RANSAC_MOTION *)aom_calloc(num_desired_motions, sizeof(RANSAC_MOTION));

   // Allocate one large buffer which will be carved up to store the inlier
   // indices for the current motion plus the num_desired_motions many
   // output models
   // This allows us to keep the allocation/deallocation logic simple, without
   // having to (for example) check that `motions` is non-null before allocating
   // the inlier arrays
   int *inlier_buffer = (int *)aom_malloc(sizeof(*inlier_buffer) * npoints *
                                          (num_desired_motions + 1));

   if (!(motions && inlier_buffer)) {
     ret_val = false;
     *mem_alloc_failed = true;
     goto finish_ransac;
   }

   // Once all our allocations are known-good, we can fill in our structures
   worst_kept_motion = motions;

   for (i = 0; i < num_desired_motions; ++i) {
     motions[i].inlier_indices = inlier_buffer + i * npoints;
   }
   memset(&current_motion, 0, sizeof(current_motion));
   current_motion.inlier_indices = inlier_buffer + num_desired_motions * npoints;

   for (int trial_count = 0; trial_count < NUM_TRIALS; trial_count++) {
     lcg_pick(npoints, minpts, indices, &seed);

     if (!model_info->find_transformation(matched_points, indices, minpts,
                                          params_this_motion)) {
       continue;
     }

     model_info->score_model(params_this_motion, matched_points, npoints,
                             &current_motion);

     if (current_motion.num_inliers < min_inliers) {
       // Reject models with too few inliers
       continue;
     }

     if (is_better_motion(&current_motion, worst_kept_motion)) {
       // This motion is better than the worst currently kept motion. Remember
       // the inlier points and sse. The parameters for each kept motion
       // will be recomputed later using only the inliers.
       worst_kept_motion->num_inliers = current_motion.num_inliers;
       worst_kept_motion->sse = current_motion.sse;

       // Rather than copying the (potentially many) inlier indices from
       // current_motion.inlier_indices to worst_kept_motion->inlier_indices,
       // we can swap the underlying pointers.
       //
       // This is okay because the next time current_motion.inlier_indices
       // is used will be in the next trial, where we ignore its previous
       // contents anyway. And both arrays will be deallocated together at the
       // end of this function, so there are no lifetime issues.
       int *tmp = worst_kept_motion->inlier_indices;
       worst_kept_motion->inlier_indices = current_motion.inlier_indices;
       current_motion.inlier_indices = tmp;

       // Determine the new worst kept motion and its num_inliers and sse.
       for (i = 0; i < num_desired_motions; ++i) {
         if (is_better_motion(worst_kept_motion, &motions[i])) {
           worst_kept_motion = &motions[i];
         }
       }
     }
   }

   // Sort the motions, best first.
   qsort(motions, num_desired_motions, sizeof(RANSAC_MOTION), compare_motions);

   // Refine each of the best N models using iterative estimation.
   //
   // The idea here is loosely based on the iterative method from
   // "Locally Optimized RANSAC" by O. Chum, J. Matas and Josef Kittler:
   // https://cmp.felk.cvut.cz/ftp/articles/matas/chum-dagm03.pdf
   //
   // However, we implement a simpler version than their proposal, and simply
   // refit the model repeatedly until the number of inliers stops increasing,
   // with a cap on the number of iterations to defend against edge cases which
   // only improve very slowly.
   for (i = 0; i < num_desired_motions; ++i) {
     if (motions[i].num_inliers <= 0) {
       // Output model has already been initialized to the identity model,
       // so just skip setup
       continue;
     }

     bool bad_model = false;
     for (int refine_count = 0; refine_count < NUM_REFINES; refine_count++) {
       int num_inliers = motions[i].num_inliers;
       assert(num_inliers >= min_inliers);

       if (!model_info->find_transformation(matched_points,
                                            motions[i].inlier_indices,
                                            num_inliers, params_this_motion)) {
         // In the unlikely event that this model fitting fails, we don't have a
         // good fallback. So leave this model set to the identity model
         bad_model = true;
         break;
       }

       // Score the newly generated model
       model_info->score_model(params_this_motion, matched_points, npoints,
                               &current_motion);

       // At this point, there are three possibilities:
       // 1) If we found more inliers, keep refining.
       // 2) If we found the same number of inliers but a lower SSE, we want to
       //    keep the new model, but further refinement is unlikely to gain much.
       //    So commit to this new model
       // 3) It is possible, but very unlikely, that the new model will have
       //    fewer inliers. If it does happen, we probably just lost a few
       //    borderline inliers. So treat the same as case (2).
       if (current_motion.num_inliers > motions[i].num_inliers) {
         motions[i].num_inliers = current_motion.num_inliers;
         motions[i].sse = current_motion.sse;
         int *tmp = motions[i].inlier_indices;
         motions[i].inlier_indices = current_motion.inlier_indices;
         current_motion.inlier_indices = tmp;
       } else {
         // Refined model is no better, so stop
         // This shouldn't be significantly worse than the previous model,
         // so it's fine to use the parameters in params_this_motion.
         // This saves us from having to cache the previous iteration's params.
         break;
       }
     }

     if (bad_model) continue;

     // Fill in output struct
     memcpy(motion_models[i].params, params_this_motion,
            MAX_PARAMDIM * sizeof(*motion_models[i].params));
     for (int j = 0; j < motions[i].num_inliers; j++) {
       int index = motions[i].inlier_indices[j];
       const Correspondence *corr = &matched_points[index];
       motion_models[i].inliers[2 * j + 0] = (int)rint(corr->x);
       motion_models[i].inliers[2 * j + 1] = (int)rint(corr->y);
     }
     motion_models[i].num_inliers = motions[i].num_inliers;
   }

 finish_ransac:
   aom_free(inlier_buffer);
   aom_free(motions);

   return ret_val;
 }

 static const RansacModelInfo ransac_model_info[TRANS_TYPES] = {
   // IDENTITY
   { NULL, NULL, 0 },
 // TRANSLATION
 #if ALLOW_TRANSLATION_MODELS
   { find_translation, score_translation, 1 },
 #else
   { NULL, NULL, 0 },
 #endif
   // ROTZOOM
   { find_rotzoom, score_affine, 2 },
   // AFFINE
   { find_affine, score_affine, 3 },
 };

 // Returns true on success, false on error
 bool ransac(const Correspondence *matched_points, int npoints,
             TransformationType type, MotionModel *motion_models,
             int num_desired_motions, bool *mem_alloc_failed) {
 #if ALLOW_TRANSLATION_MODELS
   assert(type > IDENTITY && type < TRANS_TYPES);
 #else
   assert(type > TRANSLATION && type < TRANS_TYPES);
 #endif  // ALLOW_TRANSLATION_MODELS

   return ransac_internal(matched_points, npoints, motion_models,
                          num_desired_motions, &ransac_model_info[type],
                          mem_alloc_failed);
 }
	/*
	* Copyright (c) 2016, Alliance for Open Media. All rights reserved.
	*
	* This source code is subject to the terms of the BSD 2 Clause License and
	* the Alliance for Open Media Patent License 1.0. If the BSD 2 Clause License
	* was not distributed with this source code in the LICENSE file, you can
	* obtain it at www.aomedia.org/license/software. If the Alliance for Open
	* Media Patent License 1.0 was not distributed with this source code in the
	* PATENTS file, you can obtain it at www.aomedia.org/license/patent.
	*/

	#include <memory.h>
	#include <math.h>
	#include <time.h>
	#include <stdio.h>
	#include <stdbool.h>
	#include <string.h>
	#include <assert.h>

	#include "aom_dsp/flow_estimation/ransac.h"
	#include "aom_dsp/mathutils.h"
	#include "aom_mem/aom_mem.h"

	// TODO(rachelbarker): Remove dependence on code in av1/encoder/
	#include "av1/encoder/random.h"

	#define MAX_MINPTS 4
	#define MINPTS_MULTIPLIER 5

	#define INLIER_THRESHOLD 1.25
	#define INLIER_THRESHOLD_SQUARED (INLIER_THRESHOLD * INLIER_THRESHOLD)

	// Number of initial models to generate
	#define NUM_TRIALS 20

	// Number of times to refine the best model found
	#define NUM_REFINES 5

	// Flag to enable functions for finding TRANSLATION type models.
	//
	// These modes are not considered currently due to a spec bug (see comments
	// in gm_get_motion_vector() in av1/common/mv.h). Thus we don't need to compile
	// the corresponding search functions, but it is nice to keep the source around
	// but disabled, for completeness.
	#define ALLOW_TRANSLATION_MODELS 0

	typedef struct {
	int num_inliers;
	double sse; // Sum of squared errors of inliers
	int *inlier_indices;
	} RANSAC_MOTION;

	////////////////////////////////////////////////////////////////////////////////
	// ransac
	typedef bool (FindTransformationFunc)(const Correspondence points,
	const int *indices, int num_indices,
	double *params);
	typedef void (ScoreModelFunc)(const double mat, const Correspondence *points,
	int num_points, RANSAC_MOTION *model);

	// vtable-like structure which stores all of the information needed by RANSAC
	// for a particular model type
	typedef struct {
	FindTransformationFunc find_transformation;
	ScoreModelFunc score_model;

	// The minimum number of points which can be passed to find_transformation
	// to generate a model.
	//
	// This should be set as small as possible. This is due to an observation
	// from section 4 of "Optimal Ransac" by A. Hast, J. Nysjö and
	// A. Marchetti (https://dspace5.zcu.cz/bitstream/11025/6869/1/Hast.pdf):
	// using the minimum possible number of points in the initial model maximizes
	// the chance that all of the selected points are inliers.
	//
	// That paper proposes a method which can deal with models which are
	// contaminated by outliers, which helps in cases where the inlier fraction
	// is low. However, for our purposes, global motion only gives significant
	// gains when the inlier fraction is high.
	//
	// So we do not use the method from this paper, but we do find that
	// minimizing the number of points used for initial model fitting helps
	// make the best use of the limited number of models we consider.
	int minpts;
	} RansacModelInfo;

	#if ALLOW_TRANSLATION_MODELS
	static void score_translation(const double mat, const Correspondence points,
	int num_points, RANSAC_MOTION *model) {
	model->num_inliers = 0;
	model->sse = 0.0;

	for (int i = 0; i < num_points; ++i) {
	const double x1 = points[i].x;
	const double y1 = points[i].y;
	const double x2 = points[i].rx;
	const double y2 = points[i].ry;

	const double proj_x = x1 + mat[0];
	const double proj_y = y1 + mat[1];

	const double dx = proj_x - x2;
	const double dy = proj_y - y2;
	const double sse = dx * dx + dy * dy;

	if (sse < INLIER_THRESHOLD_SQUARED) {
	model->inlier_indices[model->num_inliers++] = i;
	model->sse += sse;
	}
	}
	}
	#endif // ALLOW_TRANSLATION_MODELS

	static void score_affine(const double mat, const Correspondence points,
	int num_points, RANSAC_MOTION *model) {
	model->num_inliers = 0;
	model->sse = 0.0;

	for (int i = 0; i < num_points; ++i) {
	const double x1 = points[i].x;
	const double y1 = points[i].y;
	const double x2 = points[i].rx;
	const double y2 = points[i].ry;

	const double proj_x = mat[2] * x1 + mat[3] * y1 + mat[0];
	const double proj_y = mat[4] * x1 + mat[5] * y1 + mat[1];

	const double dx = proj_x - x2;
	const double dy = proj_y - y2;
	const double sse = dx * dx + dy * dy;

	if (sse < INLIER_THRESHOLD_SQUARED) {
	model->inlier_indices[model->num_inliers++] = i;
	model->sse += sse;
	}
	}
	}

	#if ALLOW_TRANSLATION_MODELS
	static bool find_translation(const Correspondence points, const int indices,
	int num_indices, double *params) {
	double sumx = 0;
	double sumy = 0;

	for (int i = 0; i < num_indices; ++i) {
	int index = indices[i];
	const double sx = points[index].x;
	const double sy = points[index].y;
	const double dx = points[index].rx;
	const double dy = points[index].ry;

	sumx += dx - sx;
	sumy += dy - sy;
	}

	params[0] = sumx / np;
	params[1] = sumy / np;
	params[2] = 1;
	params[3] = 0;
	params[4] = 0;
	params[5] = 1;
	return true;
	}
	#endif // ALLOW_TRANSLATION_MODELS

	static bool find_rotzoom(const Correspondence points, const int indices,
	int num_indices, double *params) {
	const int n = 4; // Size of least-squares problem
	double mat[4 * 4]; // Accumulator for A'A
	double y[4]; // Accumulator for A'b
	double a[4]; // Single row of A
	double b; // Single element of b

	least_squares_init(mat, y, n);
	for (int i = 0; i < num_indices; ++i) {
	int index = indices[i];
	const double sx = points[index].x;
	const double sy = points[index].y;
	const double dx = points[index].rx;
	const double dy = points[index].ry;

	a[0] = 1;
	a[1] = 0;
	a[2] = sx;
	a[3] = sy;
	b = dx;
	least_squares_accumulate(mat, y, a, b, n);

	a[0] = 0;
	a[1] = 1;
	a[2] = sy;
	a[3] = -sx;
	b = dy;
	least_squares_accumulate(mat, y, a, b, n);
	}

	// Fill in params[0] .. params[3] with output model
	if (!least_squares_solve(mat, y, params, n)) {
	return false;
	}

	// Fill in remaining parameters
	params[4] = -params[3];
	params[5] = params[2];

	return true;
	}

	static bool find_affine(const Correspondence points, const int indices,
	int num_indices, double *params) {
	// Note: The least squares problem for affine models is 6-dimensional,
	// but it splits into two independent 3-dimensional subproblems.
	// Solving these two subproblems separately and recombining at the end
	// results in less total computation than solving the 6-dimensional
	// problem directly.
	//
	// The two subproblems correspond to all the parameters which contribute
	// to the x output of the model, and all the parameters which contribute
	// to the y output, respectively.

	const int n = 3; // Size of each least-squares problem
	double mat[2][3 * 3]; // Accumulator for A'A
	double y[2][3]; // Accumulator for A'b
	double x[2][3]; // Output vector
	double a[2][3]; // Single row of A
	double b[2]; // Single element of b

	least_squares_init(mat[0], y[0], n);
	least_squares_init(mat[1], y[1], n);
	for (int i = 0; i < num_indices; ++i) {
	int index = indices[i];
	const double sx = points[index].x;
	const double sy = points[index].y;
	const double dx = points[index].rx;
	const double dy = points[index].ry;

	a[0][0] = 1;
	a[0][1] = sx;
	a[0][2] = sy;
	b[0] = dx;
	least_squares_accumulate(mat[0], y[0], a[0], b[0], n);

	a[1][0] = 1;
	a[1][1] = sx;
	a[1][2] = sy;
	b[1] = dy;
	least_squares_accumulate(mat[1], y[1], a[1], b[1], n);
	}

	if (!least_squares_solve(mat[0], y[0], x[0], n)) {
	return false;
	}
	if (!least_squares_solve(mat[1], y[1], x[1], n)) {
	return false;
	}

	// Rearrange least squares result to form output model
	params[0] = x[0][0];
	params[1] = x[1][0];
	params[2] = x[0][1];
	params[3] = x[0][2];
	params[4] = x[1][1];
	params[5] = x[1][2];

	return true;
	}

	// Return -1 if 'a' is a better motion, 1 if 'b' is better, 0 otherwise.
	static int compare_motions(const void arg_a, const void arg_b) {
	const RANSAC_MOTION motion_a = (RANSAC_MOTION )arg_a;
	const RANSAC_MOTION motion_b = (RANSAC_MOTION )arg_b;

	if (motion_a->num_inliers > motion_b->num_inliers) return -1;
	if (motion_a->num_inliers < motion_b->num_inliers) return 1;
	if (motion_a->sse < motion_b->sse) return -1;
	if (motion_a->sse > motion_b->sse) return 1;
	return 0;
	}

	static bool is_better_motion(const RANSAC_MOTION *motion_a,
	const RANSAC_MOTION *motion_b) {
	return compare_motions(motion_a, motion_b) < 0;
	}

	// Returns true on success, false on error
	static bool ransac_internal(const Correspondence *matched_points, int npoints,
	MotionModel *motion_models, int num_desired_motions,
	const RansacModelInfo *model_info,
	bool *mem_alloc_failed) {
	assert(npoints >= 0);
	int i = 0;
	int minpts = model_info->minpts;
	bool ret_val = true;

	unsigned int seed = (unsigned int)npoints;

	int indices[MAX_MINPTS] = { 0 };

	// Store information for the num_desired_motions best transformations found
	// and the worst motion among them, as well as the motion currently under
	// consideration.
	RANSAC_MOTION motions, worst_kept_motion = NULL;
	RANSAC_MOTION current_motion;

	// Store the parameters and the indices of the inlier points for the motion
	// currently under consideration.
	double params_this_motion[MAX_PARAMDIM];

	// Initialize output models, as a fallback in case we can't find a model
	for (i = 0; i < num_desired_motions; i++) {
	memcpy(motion_models[i].params, kIdentityParams,
	MAX_PARAMDIM * sizeof(*(motion_models[i].params)));
	motion_models[i].num_inliers = 0;
	}

	if (npoints < minpts * MINPTS_MULTIPLIER \|\| npoints == 0) {
	return false;
	}

	int min_inliers = AOMMAX((int)(MIN_INLIER_PROB * npoints), minpts);

	motions =
	(RANSAC_MOTION *)aom_calloc(num_desired_motions, sizeof(RANSAC_MOTION));

	// Allocate one large buffer which will be carved up to store the inlier
	// indices for the current motion plus the num_desired_motions many
	// output models
	// This allows us to keep the allocation/deallocation logic simple, without
	// having to (for example) check that `motions` is non-null before allocating
	// the inlier arrays
	int inlier_buffer = (int )aom_malloc(sizeof(inlier_buffer) npoints *
	(num_desired_motions + 1));

	if (!(motions && inlier_buffer)) {
	ret_val = false;
	*mem_alloc_failed = true;
	goto finish_ransac;
	}

	// Once all our allocations are known-good, we can fill in our structures
	worst_kept_motion = motions;

	for (i = 0; i < num_desired_motions; ++i) {
	motions[i].inlier_indices = inlier_buffer + i * npoints;
	}
	memset(&current_motion, 0, sizeof(current_motion));
	current_motion.inlier_indices = inlier_buffer + num_desired_motions * npoints;

	for (int trial_count = 0; trial_count < NUM_TRIALS; trial_count++) {
	lcg_pick(npoints, minpts, indices, &seed);

	if (!model_info->find_transformation(matched_points, indices, minpts,
	params_this_motion)) {
	continue;
	}

	model_info->score_model(params_this_motion, matched_points, npoints,
	&current_motion);

	if (current_motion.num_inliers < min_inliers) {
	// Reject models with too few inliers
	continue;
	}

	if (is_better_motion(&current_motion, worst_kept_motion)) {
	// This motion is better than the worst currently kept motion. Remember
	// the inlier points and sse. The parameters for each kept motion
	// will be recomputed later using only the inliers.
	worst_kept_motion->num_inliers = current_motion.num_inliers;
	worst_kept_motion->sse = current_motion.sse;

	// Rather than copying the (potentially many) inlier indices from
	// current_motion.inlier_indices to worst_kept_motion->inlier_indices,
	// we can swap the underlying pointers.
	//
	// This is okay because the next time current_motion.inlier_indices
	// is used will be in the next trial, where we ignore its previous
	// contents anyway. And both arrays will be deallocated together at the
	// end of this function, so there are no lifetime issues.
	int *tmp = worst_kept_motion->inlier_indices;
	worst_kept_motion->inlier_indices = current_motion.inlier_indices;
	current_motion.inlier_indices = tmp;

	// Determine the new worst kept motion and its num_inliers and sse.
	for (i = 0; i < num_desired_motions; ++i) {
	if (is_better_motion(worst_kept_motion, &motions[i])) {
	worst_kept_motion = &motions[i];
	}
	}
	}
	}

	// Sort the motions, best first.
	qsort(motions, num_desired_motions, sizeof(RANSAC_MOTION), compare_motions);

	// Refine each of the best N models using iterative estimation.
	//
	// The idea here is loosely based on the iterative method from
	// "Locally Optimized RANSAC" by O. Chum, J. Matas and Josef Kittler:
	// https://cmp.felk.cvut.cz/ftp/articles/matas/chum-dagm03.pdf
	//
	// However, we implement a simpler version than their proposal, and simply
	// refit the model repeatedly until the number of inliers stops increasing,
	// with a cap on the number of iterations to defend against edge cases which
	// only improve very slowly.
	for (i = 0; i < num_desired_motions; ++i) {
	if (motions[i].num_inliers <= 0) {
	// Output model has already been initialized to the identity model,
	// so just skip setup
	continue;
	}

	bool bad_model = false;
	for (int refine_count = 0; refine_count < NUM_REFINES; refine_count++) {
	int num_inliers = motions[i].num_inliers;
	assert(num_inliers >= min_inliers);

	if (!model_info->find_transformation(matched_points,
	motions[i].inlier_indices,
	num_inliers, params_this_motion)) {
	// In the unlikely event that this model fitting fails, we don't have a
	// good fallback. So leave this model set to the identity model
	bad_model = true;
	break;
	}

	// Score the newly generated model
	model_info->score_model(params_this_motion, matched_points, npoints,
	&current_motion);

	// At this point, there are three possibilities:
	// 1) If we found more inliers, keep refining.
	// 2) If we found the same number of inliers but a lower SSE, we want to
	// keep the new model, but further refinement is unlikely to gain much.
	// So commit to this new model
	// 3) It is possible, but very unlikely, that the new model will have
	// fewer inliers. If it does happen, we probably just lost a few
	// borderline inliers. So treat the same as case (2).
	if (current_motion.num_inliers > motions[i].num_inliers) {
	motions[i].num_inliers = current_motion.num_inliers;
	motions[i].sse = current_motion.sse;
	int *tmp = motions[i].inlier_indices;
	motions[i].inlier_indices = current_motion.inlier_indices;
	current_motion.inlier_indices = tmp;
	} else {
	// Refined model is no better, so stop
	// This shouldn't be significantly worse than the previous model,
	// so it's fine to use the parameters in params_this_motion.
	// This saves us from having to cache the previous iteration's params.
	break;
	}
	}

	if (bad_model) continue;

	// Fill in output struct
	memcpy(motion_models[i].params, params_this_motion,
	MAX_PARAMDIM * sizeof(*motion_models[i].params));
	for (int j = 0; j < motions[i].num_inliers; j++) {
	int index = motions[i].inlier_indices[j];
	const Correspondence *corr = &matched_points[index];
	motion_models[i].inliers[2 * j + 0] = (int)rint(corr->x);
	motion_models[i].inliers[2 * j + 1] = (int)rint(corr->y);
	}
	motion_models[i].num_inliers = motions[i].num_inliers;
	}

	finish_ransac:
	aom_free(inlier_buffer);
	aom_free(motions);

	return ret_val;
	}

	static const RansacModelInfo ransac_model_info[TRANS_TYPES] = {
	// IDENTITY
	{ NULL, NULL, 0 },
	// TRANSLATION
	#if ALLOW_TRANSLATION_MODELS
	{ find_translation, score_translation, 1 },
	#else
	{ NULL, NULL, 0 },
	#endif
	// ROTZOOM
	{ find_rotzoom, score_affine, 2 },
	// AFFINE
	{ find_affine, score_affine, 3 },
	};

	// Returns true on success, false on error
	bool ransac(const Correspondence *matched_points, int npoints,
	TransformationType type, MotionModel *motion_models,
	int num_desired_motions, bool *mem_alloc_failed) {
	#if ALLOW_TRANSLATION_MODELS
	assert(type > IDENTITY && type < TRANS_TYPES);
	#else
	assert(type > TRANSLATION && type < TRANS_TYPES);
	#endif // ALLOW_TRANSLATION_MODELS

	return ransac_internal(matched_points, npoints, motion_models,
	num_desired_motions, &ransac_model_info[type],
	mem_alloc_failed);
	}