rho.cpp

/*
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  By downloading, copying, installing or using the software you agree to this license.
  If you do not agree to this license, do not download, install,
  copy or use the software.


                          BSD 3-Clause License

 Copyright (C) 2014, Olexa Bilaniuk, Hamid Bazargani & Robert Laganiere, all rights reserved.

 Redistribution and use in source and binary forms, with or without modification,
 are permitted provided that the following conditions are met:

   * Redistribution's of source code must retain the above copyright notice,
     this list of conditions and the following disclaimer.

   * Redistribution's in binary form must reproduce the above copyright notice,
     this list of conditions and the following disclaimer in the documentation
     and/or other materials provided with the distribution.

   * The name of the copyright holders may not be used to endorse or promote products
     derived from this software without specific prior written permission.

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 and on any theory of liability, whether in contract, strict liability,
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*/

/**
 * Bilaniuk, Olexa, Hamid Bazargani, and Robert Laganiere. "Fast Target
 * Recognition on Mobile Devices: Revisiting Gaussian Elimination for the
 * Estimation of Planar Homographies." In Computer Vision and Pattern
 * Recognition Workshops (CVPRW), 2014 IEEE Conference on, pp. 119-125.
 * IEEE, 2014.
 */

/* Includes */
#include "precomp.hpp"
#include <opencv2/core.hpp>
#include <stdlib.h>
#include <stdio.h>
#include <stdint.h>
#include <string.h>
#include <stddef.h>
#include <limits.h>
#include <float.h>
#include <math.h>
#include <vector>
#include "rho.h"





/* For the sake of cv:: namespace ONLY: */
namespace cv{/* For C support, replace with extern "C" { */


/* Constants */
const int    MEM_ALIGN              = 32;
const size_t HSIZE                  = (3*3*sizeof(float));
const double MIN_DELTA_CHNG         = 0.1;
// const double CHI_STAT               = 2.706;
const double CHI_SQ                 = 1.645;
// const double RLO                    = 0.25;
// const double RHI                    = 0.75;
const int    MAXLEVMARQITERS        = 100;
const int    SMPL_SIZE              = 4;      /* 4 points required per model */
const int    SPRT_T_M               = 25;     /* Guessing 25 match evlauations / 1 model generation */
const int    SPRT_M_S               = 1;      /* 1 model per sample */
const double SPRT_EPSILON           = 0.1;    /* No explanation */
const double SPRT_DELTA             = 0.01;   /* No explanation */
const double LM_GAIN_LO             = 0.25;   /* See sacLMGain(). */
const double LM_GAIN_HI             = 0.75;   /* See sacLMGain(). */


/* Data Structures */

/**
 * Base Struct for RHO algorithm.
 *
 * A RHO estimator has initialization, finalization, capacity, seeding and
 * homography-estimation APIs that must be implemented.
 */

struct RHO_HEST{
    /* This is a virtual base class; It should have a virtual destructor. */
    virtual ~RHO_HEST(){}

    /* External Interface Methods */

    /**
     * Initialization work.
     *
     * @return 0 if initialization is unsuccessful; non-zero otherwise.
     */

    virtual inline int    initialize(void){return 1;}


    /**
     * Finalization work.
     */

    virtual inline void   finalize(void){}

    /**
     * Ensure that the estimator context's internal table for the non-randomness
     * criterion is at least of the given size, and uses the given beta. The table
     * should be larger than the maximum number of matches fed into the estimator.
     *
     * A value of N of 0 requests deallocation of the table.
     *
     * @param [in] N     If 0, deallocate internal table. If > 0, ensure that the
     *                   internal table is of at least this size, reallocating if
     *                   necessary.
     * @param [in] beta  The beta-factor to use within the table.
     * @return 0 if unsuccessful; non-zero otherwise.
     */

    virtual inline int    ensureCapacity(unsigned N, double beta){
        (void)N;
        (void)beta;

        return 1;
    }


    /**
     * Generates a random double uniformly distributed in the range [0, 1).
     *
     * The default implementation uses the xorshift128+ algorithm from
     * Sebastiano Vigna. Further scramblings of Marsaglia's xorshift generators.
     * CoRR, abs/1402.6246, 2014.
     * http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf
     *
     * Source roughly as given in
     * http://en.wikipedia.org/wiki/Xorshift#Xorshift.2B
     */

    virtual inline double fastRandom(void){
        uint64_t x = prng.s[0];
        uint64_t y = prng.s[1];
        x ^= x << 23; // a
        x ^= x >> 17; // b
        x ^= y ^ (y >> 26); // c
        prng.s[0] = y;
        prng.s[1] = x;
        uint64_t s = x + y;

        return s * 5.421010862427522e-20;/* 2^-64 */
    }


    /**
     * Seeds the context's PRNG.
     *
     * @param [in] seed  A 64-bit unsigned integer seed.
     */

    virtual inline void   fastSeed(uint64_t seed){
        int i;

        prng.s[0] =  seed;
        prng.s[1] = ~seed;/* Guarantees one of the elements will be non-zero. */

        /**
         * Escape from zero-land (see xorshift128+ paper). Approximately 20
         * iterations required according to the graph.
         */

        for(i=0;i<20;i++){
            fastRandom();
        }
    }


    /**
     * Estimates the homography using the given context, matches and parameters to
     * PROSAC.
     *
     * @param [in]     src     The pointer to the source points of the matches.
     *                             Cannot be NULL.
     * @param [in]     dst     The pointer to the destination points of the matches.
     *                             Cannot be NULL.
     * @param [out]    inl     The pointer to the output mask of inlier matches.
     *                             May be NULL.
     * @param [in]     N       The number of matches.
     * @param [in]     maxD    The maximum distance.
     * @param [in]     maxI    The maximum number of PROSAC iterations.
     * @param [in]     rConvg  The RANSAC convergence parameter.
     * @param [in]     cfd     The required confidence in the solution.
     * @param [in]     minInl  The minimum required number of inliers.
     * @param [in]     beta    The beta-parameter for the non-randomness criterion.
     * @param [in]     flags   A union of flags to control the estimation.
     * @param [in]     guessH  An extrinsic guess at the solution H, or NULL if
     *                         none provided.
     * @param [out]    finalH  The final estimation of H, or the zero matrix if
     *                         the minimum number of inliers was not met.
     *                         Cannot be NULL.
     * @return                 The number of inliers if the minimum number of
     *                         inliers for acceptance was reached; 0 otherwise.
     */

    virtual unsigned      rhoHest(const float*   src,     /* Source points */
                                  const float*   dst,     /* Destination points */
                                  char*          inl,     /* Inlier mask */
                                  unsigned       N,       /*  = src.length = dst.length = inl.length */
                                  float          maxD,    /* Works:     3.0 */
                                  unsigned       maxI,    /* Works:    2000 */
                                  unsigned       rConvg,  /* Works:    2000 */
                                  double         cfd,     /* Works:   0.995 */
                                  unsigned       minInl,  /* Minimum:     4 */
                                  double         beta,    /* Works:    0.35 */
                                  unsigned       flags,   /* Works:       0 */
                                  const float*   guessH,  /* Extrinsic guess, NULL if none provided */
                                  float*         finalH) = 0; /* Final result. */



    /* PRNG XORshift128+ */
    struct{
        uint64_t  s[2];            /* PRNG state */
    } prng;
};



/**
 * Generic C implementation of RHO algorithm.
 */

struct RHO_HEST_REFC : RHO_HEST{
    /**
     * Virtual Arguments.
     *
     * Exactly the same as at function call, except:
     * - minInl is enforced to be >= 4.
     */

    struct{
        const float* src;
        const float* dst;
        char*        inl;
        unsigned     N;
        float        maxD;
        unsigned     maxI;
        unsigned     rConvg;
        double       cfd;
        unsigned     minInl;
        double       beta;
        unsigned     flags;
        const float* guessH;
        float*       finalH;
    } arg;

    /* PROSAC Control */
    struct{
        unsigned  i;               /* Iteration Number */
        unsigned  phNum;           /* Phase Number */
        unsigned  phEndI;          /* Phase End Iteration */
        double    phEndFpI;        /* Phase floating-point End Iteration */
        unsigned  phMax;           /* Termination phase number */
        unsigned  phNumInl;        /* Number of inliers for termination phase */
        unsigned  numModels;       /* Number of models tested */
        unsigned* smpl;            /* Sample of match indexes */
    } ctrl;

    /* Current model being tested */
    struct{
        float*    pkdPts;          /* Packed points */
        float*    H;               /* Homography */
        char*     inl;             /* Mask of inliers */
        unsigned  numInl;          /* Number of inliers */
    } curr;

    /* Best model (so far) */
    struct{
        float*    H;               /* Homography */
        char*     inl;             /* Mask of inliers */
        unsigned  numInl;          /* Number of inliers */
    } best;

    /* Non-randomness criterion */
    struct{
        std::vector<unsigned> tbl; /* Non-Randomness: Table */
        unsigned  size;            /* Non-Randomness: Size */
        double    beta;            /* Non-Randomness: Beta */
    } nr;

    /* SPRT Evaluator */
    struct{
        double    t_M;             /* t_M */
        double    m_S;             /* m_S */
        double    epsilon;         /* Epsilon */
        double    delta;           /* delta */
        double    A;               /* SPRT Threshold */
        unsigned  Ntested;         /* Number of points tested */
        unsigned  Ntestedtotal;    /* Number of points tested in total */
        int       good;            /* Good/bad flag */
        double    lambdaAccept;    /* Accept multiplier */
        double    lambdaReject;    /* Reject multiplier */
    } eval;

    /* Levenberg-Marquardt Refinement */
    struct{
        float  (* JtJ)[8];         /* JtJ matrix */
        float  (* tmp1)[8];        /* Temporary 1 */
        float*    Jte;             /* Jte vector */
    } lm;

    /* Memory Management */
    struct{
        cv::Mat perObj;
        cv::Mat perRun;
    } mem;

    /* Initialized? */
    int initialized;


    /* Empty constructors and destructors */
    public:
    RHO_HEST_REFC();
    private: /* Forbid copying. */
    RHO_HEST_REFC(const RHO_HEST_REFC&);
    public:
    ~RHO_HEST_REFC();

    /* Methods to implement external interface */
    inline int    initialize(void);
    inline void   finalize(void);
    inline int    ensureCapacity(unsigned N, double beta);
    unsigned      rhoHest(const float*   src,     /* Source points */
                          const float*   dst,     /* Destination points */
                          char*          inl,     /* Inlier mask */
                          unsigned       N,       /*  = src.length = dst.length = inl.length */
                          float          maxD,    /* Works:     3.0 */
                          unsigned       maxI,    /* Works:    2000 */
                          unsigned       rConvg,  /* Works:    2000 */
                          double         cfd,     /* Works:   0.995 */
                          unsigned       minInl,  /* Minimum:     4 */
                          double         beta,    /* Works:    0.35 */
                          unsigned       flags,   /* Works:       0 */
                          const float*   guessH,  /* Extrinsic guess, NULL if none provided */
                          float*         finalH); /* Final result. */



    /* Methods to implement internals */
    inline void   allocatePerObj(void);
    inline void   allocatePerRun(void);
    inline void   deallocatePerRun(void);
    inline void   deallocatePerObj(void);
    inline int    initRun(void);
    inline void   finiRun(void);
    inline int    haveExtrinsicGuess(void);
    inline int    hypothesize(void);
    inline int    verify(void);
    inline int    isNREnabled(void);
    inline int    isRefineEnabled(void);
    inline int    isFinalRefineEnabled(void);
    inline int    PROSACPhaseEndReached(void);
    inline void   PROSACGoToNextPhase(void);
    inline void   getPROSACSample(void);
    inline void   rndSmpl(unsigned  sampleSize,
                          unsigned* currentSample,
                          unsigned  dataSetSize);
    inline int    isSampleDegenerate(void);
    inline void   generateModel(void);
    inline int    isModelDegenerate(void);
    inline void   evaluateModelSPRT(void);
    inline void   updateSPRT(void);
    inline void   designSPRTTest(void);
    inline int    isBestModel(void);
    inline int    isBestModelGoodEnough(void);
    inline void   saveBestModel(void);
    inline void   nStarOptimize(void);
    inline void   updateBounds(void);
    inline void   outputModel(void);
    inline void   outputZeroH(void);
    inline int    canRefine(void);
    inline void   refine(void);
};




/**
 * Prototypes for purely-computational code.
 */

static inline void   sacInitNonRand       (double    beta,
                                           unsigned  start,
                                           unsigned  N,
                                           unsigned* nonRandMinInl);
static inline double sacInitPEndFpI       (const unsigned ransacConvg,
                                           const unsigned n,
                                           const unsigned s);
static inline unsigned sacCalcIterBound   (double   confidence,
                                           double   inlierRate,
                                           unsigned sampleSize,
                                           unsigned maxIterBound);
static inline void   hFuncRefC            (float* packedPoints, float* H);
static inline void   sacCalcJacobianErrors(const float* H,
                                           const float* src,
                                           const float* dst,
                                           const char*  inl,
                                           unsigned     N,
                                           float     (* JtJ)[8],
                                           float*       Jte,
                                           float*       Sp);
static inline float  sacLMGain            (const float*  dH,
                                           const float*  Jte,
                                           const float   S,
                                           const float   newS,
                                           const float   lambda);
static inline int    sacChol8x8Damped     (const float (*A)[8],
                                           float         lambda,
                                           float       (*L)[8]);
static inline void   sacTRInv8x8          (const float (*L)[8],
                                           float       (*M)[8]);
static inline void   sacTRISolve8x8       (const float (*L)[8],
                                           const float*  Jte,
                                           float*        dH);
static inline void   sacSub8x1            (float*       Hout,
                                           const float* H,
                                           const float* dH);



/* Functions */

/**
 * External access to context constructor.
 *
 * @return A pointer to the context if successful; NULL if an error occured.
 */

Ptr<RHO_HEST> rhoInit(void){
    /* Select an optimized implementation of RHO here. */

#if 1
    /**
     * For now, only the generic C implementation is available. In the future,
     * SSE2/AVX/AVX2/FMA/NEON versions may be added, and they will be selected
     * depending on cv::checkHardwareSupport()'s return values.
     */

    Ptr<RHO_HEST> p = Ptr<RHO_HEST>(new RHO_HEST_REFC);
#endif

    /* Initialize it. */
    if(p){
        if(!p->initialize()){
            p.release();
        }
    }

    /* Return it. */
    return p;
}


/**
 * External access to non-randomness table resize.
 */

int  rhoEnsureCapacity(Ptr<RHO_HEST> p, unsigned N, double beta){
    return p->ensureCapacity(N, beta);
}


/**
 * Seeds the internal PRNG with the given seed.
 */

void rhoSeed(Ptr<RHO_HEST> p, uint64_t seed){
    p->fastSeed(seed);
}


/**
 * Estimates the homography using the given context, matches and parameters to
 * PROSAC.
 *
 * @param [in/out] p       The context to use for homography estimation. Must
 *                             be already initialized. Cannot be NULL.
 * @param [in]     src     The pointer to the source points of the matches.
 *                             Must be aligned to 4 bytes. Cannot be NULL.
 * @param [in]     dst     The pointer to the destination points of the matches.
 *                             Must be aligned to 16 bytes. Cannot be NULL.
 * @param [out]    inl     The pointer to the output mask of inlier matches.
 *                             Must be aligned to 16 bytes. May be NULL.
 * @param [in]     N       The number of matches.
 * @param [in]     maxD    The maximum distance.
 * @param [in]     maxI    The maximum number of PROSAC iterations.
 * @param [in]     rConvg  The RANSAC convergence parameter.
 * @param [in]     cfd     The required confidence in the solution.
 * @param [in]     minInl  The minimum required number of inliers.
 * @param [in]     beta    The beta-parameter for the non-randomness criterion.
 * @param [in]     flags   A union of flags to control the estimation.
 * @param [in]     guessH  An extrinsic guess at the solution H, or NULL if
 *                         none provided.
 * @param [out]    finalH  The final estimation of H, or the zero matrix if
 *                         the minimum number of inliers was not met.
 *                         Cannot be NULL.
 * @return                 The number of inliers if the minimum number of
 *                         inliers for acceptance was reached; 0 otherwise.
 */

unsigned rhoHest(Ptr<RHO_HEST> p,       /* Homography estimation context. */
                 const float*  src,     /* Source points */
                 const float*  dst,     /* Destination points */
                 char*         inl,     /* Inlier mask */
                 unsigned      N,       /*  = src.length = dst.length = inl.length */
                 float         maxD,    /* Works:     3.0 */
                 unsigned      maxI,    /* Works:    2000 */
                 unsigned      rConvg,  /* Works:    2000 */
                 double        cfd,     /* Works:   0.995 */
                 unsigned      minInl,  /* Minimum:     4 */
                 double        beta,    /* Works:    0.35 */
                 unsigned      flags,   /* Works:       0 */
                 const float*  guessH,  /* Extrinsic guess, NULL if none provided */
                 float*        finalH){ /* Final result. */
    return p->rhoHest(src, dst, inl, N, maxD, maxI, rConvg, cfd, minInl, beta,
                      flags, guessH, finalH);
}












/*********************** RHO_HEST_REFC implementation **********************/

/**
 * Constructor for RHO_HEST_REFC.
 *
 * Does nothing. True initialization is done by initialize().
 */

RHO_HEST_REFC::RHO_HEST_REFC() : initialized(0){
    arg.src = 0;
    arg.dst = 0;
    arg.inl = 0;
    arg.N = 0;
    arg.maxD = 0;
    arg.maxI = 0;
    arg.rConvg = 0;
    arg.cfd = 0;
    arg.minInl = 0;
    arg.beta = 0;
    arg.flags = 0;
    arg.guessH = 0;
    arg.finalH = 0;

    ctrl.i = 0;
    ctrl.phNum = 0;
    ctrl.phEndI = 0;
    ctrl.phEndFpI = 0;
    ctrl.phMax = 0;
    ctrl.phNumInl = 0;
    ctrl.numModels = 0;
    ctrl.smpl = 0;

    curr.pkdPts = 0;
    curr.H = 0;
    curr.inl = 0;
    curr.numInl = 0;

    best.H = 0;
    best.inl = 0;
    best.numInl = 0;

    nr.size = 0;
    nr.beta = 0;

    eval.t_M = 0;
    eval.m_S = 0;
    eval.epsilon = 0;
    eval.delta = 0;
    eval.A = 0;
    eval.Ntested = 0;
    eval.Ntestedtotal = 0;
    eval.good = 0;
    eval.lambdaAccept = 0;
    eval.lambdaReject = 0;

    lm.JtJ = 0;
    lm.tmp1 = 0;
    lm.Jte = 0;
}

/**
 * Private copy constructor for RHO_HEST_REFC. Disabled.
 */

RHO_HEST_REFC::RHO_HEST_REFC(const RHO_HEST_REFC&) : initialized(0){

}

/**
 * Destructor for RHO_HEST_REFC.
 */

RHO_HEST_REFC::~RHO_HEST_REFC(){
    if(initialized){
        finalize();
    }
}



/**
 * Initialize the estimator context, by allocating the aligned buffers
 * internally needed.
 *
 * Currently there are 5 per-estimator buffers:
 * - The buffer of m indexes representing a sample
 * - The buffer of 16 floats representing m matches (x,y) -> (X,Y).
 * - The buffer for the current homography
 * - The buffer for the best-so-far homography
 * - Optionally, the non-randomness criterion table
 *
 * Returns 0 if unsuccessful and non-0 otherwise.
 */

inline int    RHO_HEST_REFC::initialize(void){
    initialized = 0;


    allocatePerObj();

    curr.inl    = NULL;
    curr.numInl = 0;

    best.inl    = NULL;
    best.numInl = 0;

    nr.size     = 0;
    nr.beta     = 0.0;


    fastSeed((uint64_t)~0);


    int areAllAllocsSuccessful = !mem.perObj.empty();

    if(!areAllAllocsSuccessful){
        finalize();
    }else{
        initialized = 1;
    }

    return areAllAllocsSuccessful;
}

/**
 * Finalize.
 *
 * Finalize the estimator context, by freeing the aligned buffers used
 * internally.
 */

inline void   RHO_HEST_REFC::finalize(void){
    if(initialized){
        deallocatePerObj();

        initialized = 0;
    }
}

/**
 * Ensure that the estimator context's internal table for non-randomness
 * criterion is at least of the given size, and uses the given beta. The table
 * should be larger than the maximum number of matches fed into the estimator.
 *
 * A value of N of 0 requests deallocation of the table.
 *
 * @param [in] N     If 0, deallocate internal table. If > 0, ensure that the
 *                   internal table is of at least this size, reallocating if
 *                   necessary.
 * @param [in] beta  The beta-factor to use within the table.
 * @return 0 if unsuccessful; non-zero otherwise.
 *
 * Reads:  nr.*
 * Writes: nr.*
 */

inline int    RHO_HEST_REFC::ensureCapacity(unsigned N, double beta){
    if(N == 0){
        /* Clear. */
        nr.tbl.clear();
        nr.size = 0;
    }else if(nr.beta != beta){
        /* Beta changed. Redo all the work. */
        nr.tbl.resize(N);
        nr.beta = beta;
        sacInitNonRand(nr.beta, 0, N, &nr.tbl[0]);
        nr.size = N;
    }else if(N > nr.size){
        /* Work is partially done. Do rest of it. */
        nr.tbl.resize(N);
        sacInitNonRand(nr.beta, nr.size, N, &nr.tbl[nr.size]);
        nr.size = N;
    }else{
        /* Work is already done. Do nothing. */
    }

    return 1;
}


/**
 * Estimates the homography using the given context, matches and parameters to
 * PROSAC.
 *
 * @param [in]     src     The pointer to the source points of the matches.
 *                             Must be aligned to 4 bytes. Cannot be NULL.
 * @param [in]     dst     The pointer to the destination points of the matches.
 *                             Must be aligned to 4 bytes. Cannot be NULL.
 * @param [out]    inl     The pointer to the output mask of inlier matches.
 *                             Must be aligned to 4 bytes. May be NULL.
 * @param [in]     N       The number of matches.
 * @param [in]     maxD    The maximum distance.
 * @param [in]     maxI    The maximum number of PROSAC iterations.
 * @param [in]     rConvg  The RANSAC convergence parameter.
 * @param [in]     cfd     The required confidence in the solution.
 * @param [in]     minInl  The minimum required number of inliers.
 * @param [in]     beta    The beta-parameter for the non-randomness criterion.
 * @param [in]     flags   A union of flags to control the estimation.
 * @param [in]     guessH  An extrinsic guess at the solution H, or NULL if
 *                         none provided.
 * @param [out]    finalH  The final estimation of H, or the zero matrix if
 *                         the minimum number of inliers was not met.
 *                         Cannot be NULL.
 * @return                 The number of inliers if the minimum number of
 *                         inliers for acceptance was reached; 0 otherwise.
 */

unsigned RHO_HEST_REFC::rhoHest(const float*   src,     /* Source points */
                                const float*   dst,     /* Destination points */
                                char*          inl,     /* Inlier mask */
                                unsigned       N,       /*  = src.length = dst.length = inl.length */
                                float          maxD,    /* Works:     3.0 */
                                unsigned       maxI,    /* Works:    2000 */
                                unsigned       rConvg,  /* Works:    2000 */
                                double         cfd,     /* Works:   0.995 */
                                unsigned       minInl,  /* Minimum:     4 */
                                double         beta,    /* Works:    0.35 */
                                unsigned       flags,   /* Works:       0 */
                                const float*   guessH,  /* Extrinsic guess, NULL if none provided */
                                float*         finalH){ /* Final result. */

    /**
     * Setup
     */

    arg.src     = src;
    arg.dst     = dst;
    arg.inl     = inl;
    arg.N       = N;
    arg.maxD    = maxD;
    arg.maxI    = maxI;
    arg.rConvg  = rConvg;
    arg.cfd     = cfd;
    arg.minInl  = minInl;
    arg.beta    = beta;
    arg.flags   = flags;
    arg.guessH  = guessH;
    arg.finalH  = finalH;
    if(!initRun()){
        outputZeroH();
        finiRun();
        return 0;
    }

    /**
     * Extrinsic Guess
     */

    if(haveExtrinsicGuess()){
        verify();
    }


    /**
     * PROSAC Loop
     */

    for(ctrl.i=0; ctrl.i < arg.maxI || ctrl.i < 100; ctrl.i++){
        hypothesize() && verify();
    }


    /**
     * Teardown
     */

    if(isFinalRefineEnabled() && canRefine()){
        refine();
    }

    outputModel();
    finiRun();
    return isBestModelGoodEnough() ? best.numInl : 0;
}


/**
 * Allocate per-object dynamic storage.
 *
 * This includes aligned, fixed-size internal buffers, but excludes any buffers
 * whose size cannot be determined ahead-of-time (before the number of matches
 * is known).
 *
 * All buffer memory is allocated in one single shot, and all pointers are
 * initialized.
 */

inline void   RHO_HEST_REFC::allocatePerObj(void){
    /* We have known sizes */
    size_t ctrl_smpl_sz   = SMPL_SIZE*sizeof(*ctrl.smpl);
    size_t curr_pkdPts_sz = SMPL_SIZE*2*2*sizeof(*curr.pkdPts);
    size_t curr_H_sz      = HSIZE;
    size_t best_H_sz      = HSIZE;
    size_t lm_JtJ_sz      = 8*8*sizeof(float);
    size_t lm_tmp1_sz     = 8*8*sizeof(float);
    size_t lm_Jte_sz      = 1*8*sizeof(float);

    /* We compute offsets */
    size_t total = 0;
#define MK_OFFSET(v)                                     \
    size_t v ## _of = total;                             \
    total = alignSize(v ## _of  +  v ## _sz, MEM_ALIGN)

    MK_OFFSET(ctrl_smpl);
    MK_OFFSET(curr_pkdPts);
    MK_OFFSET(curr_H);
    MK_OFFSET(best_H);
    MK_OFFSET(lm_JtJ);
    MK_OFFSET(lm_tmp1);
    MK_OFFSET(lm_Jte);

#undef MK_OFFSET

    /* Allocate dynamic memory managed by cv::Mat */
    mem.perObj.create(1, (int)(total + MEM_ALIGN), CV_8UC1);

    /* Extract aligned pointer */
    unsigned char* ptr = alignPtr(mem.perObj.data, MEM_ALIGN);

    /* Assign pointers */
    ctrl.smpl   = (unsigned*)  (ptr + ctrl_smpl_of);
    curr.pkdPts = (float*)     (ptr + curr_pkdPts_of);
    curr.H      = (float*)     (ptr + curr_H_of);
    best.H      = (float*)     (ptr + best_H_of);
    lm.JtJ      = (float(*)[8])(ptr + lm_JtJ_of);
    lm.tmp1     = (float(*)[8])(ptr + lm_tmp1_of);
    lm.Jte      = (float*)     (ptr + lm_Jte_of);
}


/**
 * Allocate per-run dynamic storage.
 *
 * This includes storage that is proportional to the number of points, such as
 * the inlier mask.
 */

inline void   RHO_HEST_REFC::allocatePerRun(void){
    /* We have known sizes */
    size_t best_inl_sz = arg.N;
    size_t curr_inl_sz = arg.N;

    /* We compute offsets */
    size_t total = 0;
#define MK_OFFSET(v)                                     \
    size_t v ## _of = total;                             \
    total = alignSize(v ## _of  +  v ## _sz, MEM_ALIGN)

    MK_OFFSET(best_inl);
    MK_OFFSET(curr_inl);

#undef MK_OFFSET

    /* Allocate dynamic memory managed by cv::Mat */
    mem.perRun.create(1, (int)(total + MEM_ALIGN), CV_8UC1);

    /* Extract aligned pointer */
    unsigned char* ptr = alignPtr(mem.perRun.data, MEM_ALIGN);

    /* Assign pointers */
    best.inl  = (char*)(ptr + best_inl_of);
    curr.inl  = (char*)(ptr + curr_inl_of);
}


/**
 * Deallocate per-run dynamic storage.
 *
 * Undoes the work by allocatePerRun().
 */

inline void   RHO_HEST_REFC::deallocatePerRun(void){
    best.inl  = NULL;
    curr.inl  = NULL;

    mem.perRun.release();
}


/**
 * Deallocate per-object dynamic storage.
 *
 * Undoes the work by allocatePerObj().
 */

inline void   RHO_HEST_REFC::deallocatePerObj(void){
    ctrl.smpl   = NULL;
    curr.pkdPts = NULL;
    curr.H      = NULL;
    best.H      = NULL;
    lm.JtJ      = NULL;
    lm.tmp1     = NULL;
    lm.Jte      = NULL;

    mem.perObj.release();
}


/**
 * Initialize SAC for a run given its arguments.
 *
 * Performs sanity-checks and memory allocations. Also initializes the state.
 *
 * @returns 0 if per-run initialization failed at any point; non-zero
 *          otherwise.
 *
 * Reads:  arg.*, nr.*
 * Writes: curr.*, best.*, ctrl.*, eval.*
 */

inline int    RHO_HEST_REFC::initRun(void){
    /**
     * Sanitize arguments.
     *
     * Runs zeroth because these are easy-to-check errors and unambiguously
     * mean something or other.
     */

    if(!arg.src || !arg.dst){
        /* Arguments src or dst are insane, must be != NULL */
        return 0;
    }
    if(arg.N < (unsigned)SMPL_SIZE){
        /* Argument N is insane, must be >= 4. */
        return 0;
    }
    if(arg.maxD < 0){
        /* Argument maxD is insane, must be >= 0. */
        return 0;
    }
    if(arg.cfd < 0 || arg.cfd > 1){
        /* Argument cfd is insane, must be in [0, 1]. */
        return 0;
    }
    /* Clamp minInl to 4 or higher. */
    arg.minInl = arg.minInl < (unsigned)SMPL_SIZE ? SMPL_SIZE : arg.minInl;
    if(isNREnabled() && (arg.beta <= 0 || arg.beta >= 1)){
        /* Argument beta is insane, must be in (0, 1). */
        return 0;
    }
    if(!arg.finalH){
        /* Argument finalH is insane, must be != NULL */
        return 0;
    }

    /**
     * Optional NR setup.
     *
     * Runs first because it is decoupled from most other things (*) and if it
     * fails, it is easy to recover from.
     *
     * (*) The only things this code depends on is the flags argument, the nr.*
     *     substruct and the sanity-checked N and beta arguments from above.
     */

    if(isNREnabled() && !ensureCapacity(arg.N, arg.beta)){
        return 0;
    }

    /**
     * Inlier mask alloc.
     *
     * Runs second because we want to quit as fast as possible if we can't even
     * allocate the two masks.
     */

    allocatePerRun();

    memset(best.inl, 0, arg.N);
    memset(curr.inl, 0, arg.N);

    /**
     * Reset scalar per-run state.
     *
     * Runs third because there's no point in resetting/calculating a large
     * number of fields if something in the above junk failed.
     */

    ctrl.i            = 0;
    ctrl.phNum        = SMPL_SIZE;
    ctrl.phEndI       = 1;
    ctrl.phEndFpI     = sacInitPEndFpI(arg.rConvg, arg.N, SMPL_SIZE);
    ctrl.phMax        = arg.N;
    ctrl.phNumInl     = 0;
    ctrl.numModels    = 0;

    if(haveExtrinsicGuess()){
        memcpy(curr.H, arg.guessH, HSIZE);
    }else{
        memset(curr.H, 0, HSIZE);
    }
    curr.numInl       = 0;

    memset(best.H, 0, HSIZE);
    best.numInl       = 0;

    eval.Ntested      = 0;
    eval.Ntestedtotal = 0;
    eval.good         = 1;
    eval.t_M          = SPRT_T_M;
    eval.m_S          = SPRT_M_S;
    eval.epsilon      = SPRT_EPSILON;
    eval.delta        = SPRT_DELTA;
    designSPRTTest();

    return 1;
}

/**
 * Finalize SAC run.
 *
 * Deallocates per-run allocatable resources. Currently this consists only of
 * the best and current inlier masks, which are equal in size to p->arg.N
 * bytes.
 *
 * Reads:  arg.bestInl, curr.inl, best.inl
 * Writes: curr.inl, best.inl
 */

inline void   RHO_HEST_REFC::finiRun(void){
    deallocatePerRun();
}

/**
 * Hypothesize a model.
 *
 * Selects randomly a sample (within the rules of PROSAC) and generates a
 * new current model, and applies degeneracy tests to it.
 *
 * @returns 0 if hypothesized model could be rejected early as degenerate, and
 * non-zero otherwise.
 */

inline int    RHO_HEST_REFC::hypothesize(void){
    if(PROSACPhaseEndReached()){
        PROSACGoToNextPhase();
    }

    getPROSACSample();
    if(isSampleDegenerate()){
        return 0;
    }

    generateModel();
    if(isModelDegenerate()){
        return 0;
    }

    return 1;
}

/**
 * Verify the hypothesized model.
 *
 * Given the current model, evaluate its quality. If it is better than
 * everything before, save as new best model (and possibly refine it).
 *
 * Returns 1.
 */

inline int    RHO_HEST_REFC::verify(void){
    evaluateModelSPRT();
    updateSPRT();

    if(isBestModel()){
        saveBestModel();

        if(isRefineEnabled() && canRefine()){
            refine();
        }

        updateBounds();

        if(isNREnabled()){
            nStarOptimize();
        }
    }

    return 1;
}

/**
 * Check whether extrinsic guess was provided or not.
 *
 * @return Zero if no extrinsic guess was provided; non-zero otherwiseEE.
 */

inline int    RHO_HEST_REFC::haveExtrinsicGuess(void){
    return !!arg.guessH;
}

/**
 * Check whether non-randomness criterion is enabled.
 *
 * @return Zero if non-randomness criterion disabled; non-zero if not.
 */

inline int    RHO_HEST_REFC::isNREnabled(void){
    return arg.flags & RHO_FLAG_ENABLE_NR;
}

/**
 * Check whether best-model-so-far refinement is enabled.
 *
 * @return Zero if best-model-so-far refinement disabled; non-zero if not.
 */

inline int    RHO_HEST_REFC::isRefineEnabled(void){
    return arg.flags & RHO_FLAG_ENABLE_REFINEMENT;
}

/**
 * Check whether final-model refinement is enabled.
 *
 * @return Zero if final-model refinement disabled; non-zero if not.
 */

inline int    RHO_HEST_REFC::isFinalRefineEnabled(void){
    return arg.flags & RHO_FLAG_ENABLE_FINAL_REFINEMENT;
}

/**
 * Computes whether the end of the current PROSAC phase has been reached. At
 * PROSAC phase phNum, only matches [0, phNum) are sampled from.
 *
 * Reads    (direct): ctrl.i, ctrl.phEndI, ctrl.phNum, ctrl.phMax
 * Reads   (callees): None.
 * Writes   (direct): None.
 * Writes  (callees): None.
 */

inline int    RHO_HEST_REFC::PROSACPhaseEndReached(void){
    return ctrl.i >= ctrl.phEndI && ctrl.phNum < ctrl.phMax;
}

/**
 * Updates unconditionally the necessary fields to move to the next PROSAC
 * stage.
 *
 * Not idempotent.
 *
 * Reads    (direct): ctrl.phNum, ctrl.phEndFpI, ctrl.phEndI
 * Reads   (callees): None.
 * Writes   (direct): ctrl.phNum, ctrl.phEndFpI, ctrl.phEndI
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::PROSACGoToNextPhase(void){
    double next;

    ctrl.phNum++;
    next = (ctrl.phEndFpI * ctrl.phNum)/(ctrl.phNum - SMPL_SIZE);
    ctrl.phEndI  += (unsigned)ceil(next - ctrl.phEndFpI);
    ctrl.phEndFpI = next;
}

/**
 * Get a sample according to PROSAC rules. Namely:
 * - If we're past the phase end interation, select randomly 4 out of the first
 *   phNum matches.
 * - Otherwise, select match phNum-1 and select randomly the 3 others out of
 *   the first phNum-1 matches.
 *
 * Reads    (direct): ctrl.i, ctrl.phEndI, ctrl.phNum
 * Reads   (callees): prng.s
 * Writes   (direct): ctrl.smpl
 * Writes  (callees): prng.s
 */

inline void   RHO_HEST_REFC::getPROSACSample(void){
    if(ctrl.i > ctrl.phEndI){
        /* FIXME: Dubious. Review. */
        rndSmpl(4, ctrl.smpl, ctrl.phNum);/* Used to be phMax */
    }else{
        rndSmpl(3, ctrl.smpl, ctrl.phNum-1);
        ctrl.smpl[3] = ctrl.phNum-1;
    }
}

/**
 * Choose, without repetition, sampleSize integers in the range [0, numDataPoints).
 *
 * Reads    (direct): None.
 * Reads   (callees): prng.s
 * Writes   (direct): None.
 * Writes  (callees): prng.s
 */

inline void   RHO_HEST_REFC::rndSmpl(unsigned  sampleSize,
                                     unsigned* currentSample,
                                     unsigned  dataSetSize){
    /**
     * If sampleSize is very close to dataSetSize, we use selection sampling.
     * Otherwise we use the naive sampling technique wherein we select random
     * indexes until sampleSize of them are distinct.
     */

    if(sampleSize*2>dataSetSize){
        /**
         * Selection Sampling:
         *
         * Algorithm S (Selection sampling technique). To select n records at random from a set of N, where 0 < n <= N.
         * S1. [Initialize.] Set t = 0, m = 0. (During this algorithm, m represents the number of records selected so far,
         *                                      and t is the total number of input records that we have dealt with.)
         * S2. [Generate U.] Generate a random number U, uniformly distributed between zero and one.
         * S3. [Test.] If (N - t)U >= n - m, go to step S5.
         * S4. [Select.] Select the next record for the sample, and increase m and t by 1. If m < n, go to step S2;
         *               otherwise the sample is complete and the algorithm terminates.
         * S5. [Skip.] Skip the next record (do not include it in the sample), increase t by 1, and go back to step S2.
         *
         * Replaced m with i and t with j in the below code.
         */

        unsigned i=0,j=0;

        for(i=0;i<sampleSize;j++){
            double U=fastRandom();
            if((dataSetSize-j)*U < (sampleSize-i)){
                currentSample[i++]=j;
            }
        }
    }else{
        /**
         * Naive sampling technique. Generate indexes until sampleSize of them are distinct.
         */

        unsigned i, j;
        for(i=0;i<sampleSize;i++){
            int inList;

            do{
                currentSample[i] = (unsigned)(dataSetSize*fastRandom());

                inList=0;
                for(j=0;j<i;j++){
                    if(currentSample[i] == currentSample[j]){
                        inList=1;
                        break;
                    }
                }
            }while(inList);
        }
    }
}

/**
 * Checks whether the *sample* is degenerate prior to model generation.
 * - First, the extremely cheap numerical degeneracy test is run, which weeds
 *   out bad samples to the optimized GE implementation.
 * - Second, the geometrical degeneracy test is run, which weeds out most other
 *   bad samples.
 *
 * Reads    (direct): ctrl.smpl, arg.src, arg.dst
 * Reads   (callees): None.
 * Writes   (direct): curr.pkdPts
 * Writes  (callees): None.
 */

inline int    RHO_HEST_REFC::isSampleDegenerate(void){
    unsigned i0 = ctrl.smpl[0], i1 = ctrl.smpl[1], i2 = ctrl.smpl[2], i3 = ctrl.smpl[3];
    typedef struct{float x,y;} MyPt2f;
    MyPt2f* pkdPts = (MyPt2f*)curr.pkdPts, *src = (MyPt2f*)arg.src, *dst = (MyPt2f*)arg.dst;

    /**
     * Pack the matches selected by the SAC algorithm.
     * Must be packed  points[0:7]  = {srcx0, srcy0, srcx1, srcy1, srcx2, srcy2, srcx3, srcy3}
     *                 points[8:15] = {dstx0, dsty0, dstx1, dsty1, dstx2, dsty2, dstx3, dsty3}
     * Gather 4 points into the vector
     */

    pkdPts[0] = src[i0];
    pkdPts[1] = src[i1];
    pkdPts[2] = src[i2];
    pkdPts[3] = src[i3];
    pkdPts[4] = dst[i0];
    pkdPts[5] = dst[i1];
    pkdPts[6] = dst[i2];
    pkdPts[7] = dst[i3];

    /**
     * If the matches' source points have common x and y coordinates, abort.
     */

    if(pkdPts[0].x == pkdPts[1].x || pkdPts[1].x == pkdPts[2].x ||
       pkdPts[2].x == pkdPts[3].x || pkdPts[0].x == pkdPts[2].x ||
       pkdPts[1].x == pkdPts[3].x || pkdPts[0].x == pkdPts[3].x ||
       pkdPts[0].y == pkdPts[1].y || pkdPts[1].y == pkdPts[2].y ||
       pkdPts[2].y == pkdPts[3].y || pkdPts[0].y == pkdPts[2].y ||
       pkdPts[1].y == pkdPts[3].y || pkdPts[0].y == pkdPts[3].y){
        return 1;
    }

    /* If the matches do not satisfy the strong geometric constraint, abort. */
    /* (0 x 1) * 2 */
    float cross0s0 = pkdPts[0].y-pkdPts[1].y;
    float cross0s1 = pkdPts[1].x-pkdPts[0].x;
    float cross0s2 = pkdPts[0].x*pkdPts[1].y-pkdPts[0].y*pkdPts[1].x;
    float dots0    = cross0s0*pkdPts[2].x + cross0s1*pkdPts[2].y + cross0s2;
    float cross0d0 = pkdPts[4].y-pkdPts[5].y;
    float cross0d1 = pkdPts[5].x-pkdPts[4].x;
    float cross0d2 = pkdPts[4].x*pkdPts[5].y-pkdPts[4].y*pkdPts[5].x;
    float dotd0    = cross0d0*pkdPts[6].x + cross0d1*pkdPts[6].y + cross0d2;
    if(((int)dots0^(int)dotd0) < 0){
        return 1;
    }
    /* (0 x 1) * 3 */
    float cross1s0 = cross0s0;
    float cross1s1 = cross0s1;
    float cross1s2 = cross0s2;
    float dots1    = cross1s0*pkdPts[3].x + cross1s1*pkdPts[3].y + cross1s2;
    float cross1d0 = cross0d0;
    float cross1d1 = cross0d1;
    float cross1d2 = cross0d2;
    float dotd1    = cross1d0*pkdPts[7].x + cross1d1*pkdPts[7].y + cross1d2;
    if(((int)dots1^(int)dotd1) < 0){
        return 1;
    }
    /* (2 x 3) * 0 */
    float cross2s0 = pkdPts[2].y-pkdPts[3].y;
    float cross2s1 = pkdPts[3].x-pkdPts[2].x;
    float cross2s2 = pkdPts[2].x*pkdPts[3].y-pkdPts[2].y*pkdPts[3].x;
    float dots2    = cross2s0*pkdPts[0].x + cross2s1*pkdPts[0].y + cross2s2;
    float cross2d0 = pkdPts[6].y-pkdPts[7].y;
    float cross2d1 = pkdPts[7].x-pkdPts[6].x;
    float cross2d2 = pkdPts[6].x*pkdPts[7].y-pkdPts[6].y*pkdPts[7].x;
    float dotd2    = cross2d0*pkdPts[4].x + cross2d1*pkdPts[4].y + cross2d2;
    if(((int)dots2^(int)dotd2) < 0){
        return 1;
    }
    /* (2 x 3) * 1 */
    float cross3s0 = cross2s0;
    float cross3s1 = cross2s1;
    float cross3s2 = cross2s2;
    float dots3    = cross3s0*pkdPts[1].x + cross3s1*pkdPts[1].y + cross3s2;
    float cross3d0 = cross2d0;
    float cross3d1 = cross2d1;
    float cross3d2 = cross2d2;
    float dotd3    = cross3d0*pkdPts[5].x + cross3d1*pkdPts[5].y + cross3d2;
    if(((int)dots3^(int)dotd3) < 0){
        return 1;
    }

    /* Otherwise, accept */
    return 0;
}

/**
 * Compute homography of matches in gathered, packed sample and output the
 * current homography.
 *
 * Reads    (direct): None.
 * Reads   (callees): curr.pkdPts
 * Writes   (direct): None.
 * Writes  (callees): curr.H
 */

inline void   RHO_HEST_REFC::generateModel(void){
    hFuncRefC(curr.pkdPts, curr.H);
}

/**
 * Checks whether the model is itself degenerate.
 * - One test: All elements of the homography are added, and if the result is
 *   NaN the homography is rejected.
 *
 * Reads    (direct): curr.H
 * Reads   (callees): None.
 * Writes   (direct): None.
 * Writes  (callees): None.
 */

inline int    RHO_HEST_REFC::isModelDegenerate(void){
    int degenerate;
    float* H = curr.H;
    float f=H[0]+H[1]+H[2]+H[3]+H[4]+H[5]+H[6]+H[7];

    /* degenerate = isnan(f); */
    /* degenerate = f!=f;// Only NaN is not equal to itself. */
    degenerate = cvIsNaN(f);
    /* degenerate = 0; */


    return degenerate;
}

/**
 * Evaluates the current model using SPRT for early exiting.
 *
 * Reads    (direct): arg.maxD, arg.src, arg.dst, arg.N, curr.inl, curr.H,
 *                    ctrl.numModels, eval.Ntestedtotal, eval.lambdaAccept,
 *                    eval.lambdaReject, eval.A
 * Reads   (callees): None.
 * Writes   (direct): ctrl.numModels, curr.numInl, eval.Ntested, eval.good,
 *                    eval.Ntestedtotal
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::evaluateModelSPRT(void){
    unsigned i;
    unsigned isInlier;
    double   lambda  = 1.0;
    float    distSq  = arg.maxD*arg.maxD;
    const float* src = arg.src;
    const float* dst = arg.dst;
    char*    inl     = curr.inl;
    const float*   H = curr.H;


    ctrl.numModels++;

    curr.numInl   = 0;
    eval.Ntested  = 0;
    eval.good     = 1;


    /* SCALAR */
    for(i=0;i<arg.N && eval.good;i++){
        /* Backproject */
        float x=src[i*2],y=src[i*2+1];
        float X=dst[i*2],Y=dst[i*2+1];

        float reprojX=H[0]*x+H[1]*y+H[2]; /*  ( X_1 )     ( H_11 H_12    H_13  ) (x_1)       */
        float reprojY=H[3]*x+H[4]*y+H[5]; /*  ( X_2 )  =  ( H_21 H_22    H_23  ) (x_2)       */
        float reprojZ=H[6]*x+H[7]*y+1.0f; /*  ( X_3 )     ( H_31 H_32 H_33=1.0 ) (x_3 = 1.0) */

        /* reproj is in homogeneous coordinates. To bring back to "regular" coordinates, divide by Z. */
        reprojX/=reprojZ;
        reprojY/=reprojZ;

        /* Compute distance */
        reprojX-=X;
        reprojY-=Y;
        reprojX*=reprojX;
        reprojY*=reprojY;
        float reprojDist = reprojX+reprojY;

        /* ... */
        isInlier   = reprojDist <= distSq;
        curr.numInl += isInlier;
        *inl++     = (char)isInlier;


        /* SPRT */
        lambda *= isInlier ? eval.lambdaAccept : eval.lambdaReject;
        eval.good = lambda <= eval.A;
        /* If !good, the threshold A was exceeded, so we're rejecting */
    }


    eval.Ntested       = i;
    eval.Ntestedtotal += i;
}

/**
 * Update either the delta or epsilon SPRT parameters, depending on the events
 * that transpired in the previous evaluation.
 *
 * Reads    (direct): eval.good, curr.numInl, arg.N, eval.Ntested, eval.delta
 * Reads   (callees): eval.delta, eval.epsilon, eval.t_M, eval.m_S
 * Writes   (direct): eval.epsilon, eval.delta
 * Writes  (callees): eval.A, eval.lambdaReject, eval.lambdaAccept
 */

inline void   RHO_HEST_REFC::updateSPRT(void){
    if(eval.good){
        if(isBestModel()){
            eval.epsilon = (double)curr.numInl/arg.N;
            designSPRTTest();
        }
    }else{
        double newDelta = (double)curr.numInl/eval.Ntested;

        if(newDelta > 0){
            double relChange = fabs(eval.delta - newDelta)/ eval.delta;
            if(relChange > MIN_DELTA_CHNG){
                eval.delta = newDelta;
                designSPRTTest();
            }
        }
    }
}

/**
 * Numerically compute threshold A from the estimated delta, epsilon, t_M and
 * m_S values.
 *
 * Epsilon:  Denotes the probability that a randomly chosen data point is
 *           consistent with a good model.
 * Delta:    Denotes the probability that a randomly chosen data point is
 *           consistent with a bad model.
 * t_M:      Time needed to instantiate a model hypotheses given a sample.
 *           (Computing model parameters from a sample takes the same time
 *            as verification of t_M data points)
 * m_S:      The number of models that are verified per sample.
 */

static inline double sacDesignSPRTTest(double delta, double epsilon, double t_M, double m_S){
    double An, C, K, prevAn;
    unsigned i;

    /**
     * Randomized RANSAC with Sequential Probability Ratio Test, ICCV 2005
     * Eq (2)
     */

    C = (1-delta)  *  log((1-delta)/(1-epsilon)) +
        delta      *  log(  delta  /  epsilon  );

    /**
     * Randomized RANSAC with Sequential Probability Ratio Test, ICCV 2005
     * Eq (6)
     * K = K_1/K_2 + 1 = (t_M*C)/m_S + 1
     */

    K = t_M*C/m_S + 1;

    /**
     * Randomized RANSAC with Sequential Probability Ratio Test, ICCV 2005
     * Paragraph below Eq (6)
     *
     * A* = lim_{n -> infty} A_n, where
     *     A_0     = K1/K2 + 1             and
     *     A_{n+1} = K1/K2 + 1 + log(A_n)
     * The series converges fast, typically within four iterations.
     */

    An = K;
    i  = 0;

    do{
        prevAn = An;
        An = K + log(An);
    }while((An-prevAn > 1.5e-8)  &&  (++i < 10));

    /**
     * Return A = An_stopping, with n_stopping < 10
     */

    return An;
}

/**
 * Design the SPRT test. Shorthand for
 *     A = sprt(delta, epsilon, t_M, m_S);
 *
 * Idempotent.
 *
 * Reads    (direct): eval.delta, eval.epsilon, eval.t_M, eval.m_S
 * Reads   (callees): None.
 * Writes   (direct): eval.A, eval.lambdaReject, eval.lambdaAccept.
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::designSPRTTest(void){
    eval.A = sacDesignSPRTTest(eval.delta, eval.epsilon, eval.t_M, eval.m_S);
    eval.lambdaReject = ((1.0 - eval.delta) / (1.0 - eval.epsilon));
    eval.lambdaAccept = ((   eval.delta   ) / (    eval.epsilon  ));
}

/**
 * Return whether the current model is the best model so far.
 *
 * @return Non-zero if this is the model with the most inliers seen so far;
 *         0 otherwise.
 *
 * Reads    (direct): curr.numInl, best.numInl
 * Reads   (callees): None.
 * Writes   (direct): None.
 * Writes  (callees): None.
 */

inline int    RHO_HEST_REFC::isBestModel(void){
    return curr.numInl > best.numInl;
}

/**
 * Returns whether the current-best model is good enough to be an
 * acceptable best model, by checking whether it meets the minimum
 * number of inliers.
 *
 * @return Non-zero if the current model is "good enough" to save;
 *         0 otherwise.
 *
 * Reads    (direct): best.numInl, arg.minInl
 * Reads   (callees): None.
 * Writes   (direct): None.
 * Writes  (callees): None.
 */

inline int    RHO_HEST_REFC::isBestModelGoodEnough(void){
    return best.numInl >= arg.minInl;
}

/**
 * Make current model new best model by swapping the homography, inlier mask
 * and count of inliers between the current and best models.
 *
 * Reads    (direct): curr.H, curr.inl, curr.numInl,
 *                    best.H, best.inl, best.numInl
 * Reads   (callees): None.
 * Writes   (direct): curr.H, curr.inl, curr.numInl,
 *                    best.H, best.inl, best.numInl
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::saveBestModel(void){
    float*   H      = curr.H;
    char*    inl    = curr.inl;
    unsigned numInl = curr.numInl;

    curr.H       = best.H;
    curr.inl     = best.inl;
    curr.numInl  = best.numInl;

    best.H       = H;
    best.inl     = inl;
    best.numInl  = numInl;
}

/**
 * Compute NR table entries [start, N) for given beta.
 */

static inline void   sacInitNonRand(double    beta,
                                    unsigned  start,
                                    unsigned  N,
                                    unsigned* nonRandMinInl){
    unsigned n = SMPL_SIZE+1 > start ? SMPL_SIZE+1 : start;
    double   beta_beta1_sq_chi = sqrt(beta*(1.0-beta)) * CHI_SQ;

    for(; n < N; n++){
        double   mu      = n * beta;
        double   sigma   = sqrt((double)n)* beta_beta1_sq_chi;
        unsigned i_min   = (unsigned)ceil(SMPL_SIZE + mu + sigma);

        nonRandMinInl[n] = i_min;
    }
}

/**
 * Optimize the stopping criterion to account for the non-randomness criterion
 * of PROSAC.
 *
 * Reads    (direct): arg.N, best.numInl, nr.tbl, arg.inl, ctrl.phMax,
 *                    ctrl.phNumInl, arg.cfd, arg.maxI
 * Reads   (callees): None.
 * Writes   (direct): arg.maxI, ctrl.phMax, ctrl.phNumInl
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::nStarOptimize(void){
    unsigned min_sample_length = 10*2; /*(N * INLIERS_RATIO) */
    unsigned best_n       = arg.N;
    unsigned test_n       = best_n;
    unsigned bestNumInl   = best.numInl;
    unsigned testNumInl   = bestNumInl;

    for(;test_n > min_sample_length && testNumInl;test_n--){
        if(testNumInl*best_n > bestNumInl*test_n){
            if(testNumInl < nr.tbl[test_n]){
                break;
            }
            best_n      = test_n;
            bestNumInl  = testNumInl;
        }
        testNumInl -= !!best.inl[test_n-1];
    }

    if(bestNumInl*ctrl.phMax > ctrl.phNumInl*best_n){
        ctrl.phMax    = best_n;
        ctrl.phNumInl = bestNumInl;
        arg.maxI      = sacCalcIterBound(arg.cfd,
                                         (double)ctrl.phNumInl/ctrl.phMax,
                                         SMPL_SIZE,
                                         arg.maxI);
    }
}

/**
 * Classic RANSAC iteration bound based on largest # of inliers.
 *
 * Reads    (direct): arg.maxI, arg.cfd, best.numInl, arg.N
 * Reads   (callees): None.
 * Writes   (direct): arg.maxI
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::updateBounds(void){
    arg.maxI = sacCalcIterBound(arg.cfd,
                                (double)best.numInl/arg.N,
                                SMPL_SIZE,
                                arg.maxI);
}

/**
 * Ouput the best model so far to the output argument.
 *
 * Reads    (direct): arg.finalH, best.H, arg.inl, best.inl, arg.N
 * Reads   (callees): arg.finalH, arg.inl, arg.N
 * Writes   (direct): arg.finalH, arg.inl
 * Writes  (callees): arg.finalH, arg.inl
 */

inline void   RHO_HEST_REFC::outputModel(void){
    if(isBestModelGoodEnough()){
        memcpy(arg.finalH, best.H, HSIZE);
        if(arg.inl){
            memcpy(arg.inl, best.inl, arg.N);
        }
    }else{
        outputZeroH();
    }
}

/**
 * Ouput a zeroed H to the output argument.
 *
 * Reads    (direct): arg.finalH, arg.inl, arg.N
 * Reads   (callees): None.
 * Writes   (direct): arg.finalH, arg.inl
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::outputZeroH(void){
    if(arg.finalH){
        memset(arg.finalH, 0, HSIZE);
    }
    if(arg.inl){
        memset(arg.inl,    0, arg.N);
    }
}

/**
 * Compute the real-valued number of samples per phase, given the RANSAC convergence speed,
 * data set size and sample size.
 */

static inline double sacInitPEndFpI(const unsigned ransacConvg,
                                    const unsigned n,
                                    const unsigned s){
    double numer=1, denom=1;

    unsigned i;
    for(i=0;i<s;i++){
        numer *= s-i;
        denom *= n-i;
    }

    return ransacConvg*numer/denom;
}

/**
 * Estimate the number of iterations required based on the requested confidence,
 * proportion of inliers in the best model so far and sample size.
 *
 * Clamp return value at maxIterationBound.
 */

static inline unsigned sacCalcIterBound(double   confidence,
                                        double   inlierRate,
                                        unsigned sampleSize,
                                        unsigned maxIterBound){
    unsigned retVal;

    /**
     * Formula chosen from http://en.wikipedia.org/wiki/RANSAC#The_parameters :
     *
     * \[ k = \frac{\log{(1-confidence)}}{\log{(1-inlierRate**sampleSize)}} \]
     */

    double atLeastOneOutlierProbability = 1.-pow(inlierRate, (double)sampleSize);

    /**
     * There are two special cases: When argument to log() is 0 and when it is 1.
     * Each has a special meaning.
     */

    if(atLeastOneOutlierProbability>=1.){
        /**
         * A certainty of picking at least one outlier means that we will need
         * an infinite amount of iterations in order to find a correct solution.
         */

        retVal = maxIterBound;
    }else if(atLeastOneOutlierProbability<=0.){
        /**
         * The certainty of NOT picking an outlier means that only 1 iteration
         * is needed to find a solution.
         */

        retVal = 1;
    }else{
        /**
         * Since 1-confidence (the probability of the model being based on at
         * least one outlier in the data) is equal to
         * (1-inlierRate**sampleSize)**numIterations (the probability of picking
         * at least one outlier in numIterations samples), we can isolate
         * numIterations (the return value) into
         */

        retVal = (unsigned)ceil(log(1.-confidence)/log(atLeastOneOutlierProbability));
    }

    /**
     * Clamp to maxIterationBound.
     */

    return retVal <= maxIterBound ? retVal : maxIterBound;
}


/**
 * Given 4 matches, computes the homography that relates them using Gaussian
 * Elimination. The row operations are as given in the paper.
 *
 * TODO: Clean this up. The code is hideous, and might even conceal sign bugs
 *       (specifically relating to whether the last column should be negated,
 *        or not).
 */

static void hFuncRefC(float* packedPoints,/* Source (four x,y float coordinates) points followed by
                                             destination (four x,y float coordinates) points, aligned by 32 bytes */
                      float* H){          /* Homography (three 16-byte aligned rows of 3 floats) */
    float x0=*packedPoints++;
    float y0=*packedPoints++;
    float x1=*packedPoints++;
    float y1=*packedPoints++;
    float x2=*packedPoints++;
    float y2=*packedPoints++;
    float x3=*packedPoints++;
    float y3=*packedPoints++;
    float X0=*packedPoints++;
    float Y0=*packedPoints++;
    float X1=*packedPoints++;
    float Y1=*packedPoints++;
    float X2=*packedPoints++;
    float Y2=*packedPoints++;
    float X3=*packedPoints++;
    float Y3=*packedPoints++;

    float x0X0=x0*X0, x1X1=x1*X1, x2X2=x2*X2, x3X3=x3*X3;
    float x0Y0=x0*Y0, x1Y1=x1*Y1, x2Y2=x2*Y2, x3Y3=x3*Y3;
    float y0X0=y0*X0, y1X1=y1*X1, y2X2=y2*X2, y3X3=y3*X3;
    float y0Y0=y0*Y0, y1Y1=y1*Y1, y2Y2=y2*Y2, y3Y3=y3*Y3;


    /**
     *  [0]   [1] Hidden   Prec
     *  x0    y0    1       x1
     *  x1    y1    1       x1
     *  x2    y2    1       x1
     *  x3    y3    1       x1
     *
     * Eliminate ones in column 2 and 5.
     * R(0)-=R(2)
     * R(1)-=R(2)
     * R(3)-=R(2)
     *
     *  [0]   [1] Hidden   Prec
     * x0-x2 y0-y2  0       x1+1
     * x1-x2 y1-y2  0       x1+1
     *  x2    y2    1       x1
     * x3-x2 y3-y2  0       x1+1
     *
     * Eliminate column 0 of rows 1 and 3
     * R(1)=(x0-x2)*R(1)-(x1-x2)*R(0),     y1'=(y1-y2)(x0-x2)-(x1-x2)(y0-y2)
     * R(3)=(x0-x2)*R(3)-(x3-x2)*R(0),     y3'=(y3-y2)(x0-x2)-(x3-x2)(y0-y2)
     *
     *  [0]   [1] Hidden   Prec
     * x0-x2 y0-y2  0      x1+1
     *   0    y1'   0      x2+3
     *  x2    y2    1       x1
     *   0    y3'   0      x2+3
     *
     * Eliminate column 1 of rows 0 and 3
     * R(3)=y1'*R(3)-y3'*R(1)
     * R(0)=y1'*R(0)-(y0-y2)*R(1)
     *
     *  [0]   [1] Hidden   Prec
     *  x0'    0    0      x3+5
     *   0    y1'   0      x2+3
     *  x2    y2    1       x1
     *   0     0    0      x4+7
     *
     * Eliminate columns 0 and 1 of row 2
     * R(0)/=x0'
     * R(1)/=y1'
     * R(2)-= (x2*R(0) + y2*R(1))
     *
     *  [0]   [1] Hidden   Prec
     *   1     0    0      x6+10
     *   0     1    0      x4+6
     *   0     0    1      x4+7
     *   0     0    0      x4+7
     */

    /**
     * Eliminate ones in column 2 and 5.
     * R(0)-=R(2)
     * R(1)-=R(2)
     * R(3)-=R(2)
     */

    /*float minor[4][2] = {{x0-x2,y0-y2},
                         {x1-x2,y1-y2},
                         {x2   ,y2   },
                         {x3-x2,y3-y2}};*/
    /*float major[8][3] = {{x2X2-x0X0,y2X2-y0X0,(X0-X2)},
                         {x2X2-x1X1,y2X2-y1X1,(X1-X2)},
                         {-x2X2    ,-y2X2    ,(X2   )},
                         {x2X2-x3X3,y2X2-y3X3,(X3-X2)},
                         {x2Y2-x0Y0,y2Y2-y0Y0,(Y0-Y2)},
                         {x2Y2-x1Y1,y2Y2-y1Y1,(Y1-Y2)},
                         {-x2Y2    ,-y2Y2    ,(Y2   )},
                         {x2Y2-x3Y3,y2Y2-y3Y3,(Y3-Y2)}};*/
    float minor[2][4] = {{x0-x2,x1-x2,x2   ,x3-x2},
                         {y0-y2,y1-y2,y2   ,y3-y2}};
    float major[3][8] = {{x2X2-x0X0,x2X2-x1X1,-x2X2    ,x2X2-x3X3,x2Y2-x0Y0,x2Y2-x1Y1,-x2Y2    ,x2Y2-x3Y3},
                         {y2X2-y0X0,y2X2-y1X1,-y2X2    ,y2X2-y3X3,y2Y2-y0Y0,y2Y2-y1Y1,-y2Y2    ,y2Y2-y3Y3},
                         { (X0-X2) , (X1-X2) , (X2   ) , (X3-X2) , (Y0-Y2) , (Y1-Y2) , (Y2   ) , (Y3-Y2) }};

    /**
     * int i;
     * for(i=0;i<8;i++) major[2][i]=-major[2][i];
     * Eliminate column 0 of rows 1 and 3
     * R(1)=(x0-x2)*R(1)-(x1-x2)*R(0),     y1'=(y1-y2)(x0-x2)-(x1-x2)(y0-y2)
     * R(3)=(x0-x2)*R(3)-(x3-x2)*R(0),     y3'=(y3-y2)(x0-x2)-(x3-x2)(y0-y2)
     */

    float scalar1=minor[0][0], scalar2=minor[0][1];
    minor[1][1]=minor[1][1]*scalar1-minor[1][0]*scalar2;

    major[0][1]=major[0][1]*scalar1-major[0][0]*scalar2;
    major[1][1]=major[1][1]*scalar1-major[1][0]*scalar2;
    major[2][1]=major[2][1]*scalar1-major[2][0]*scalar2;

    major[0][5]=major[0][5]*scalar1-major[0][4]*scalar2;
    major[1][5]=major[1][5]*scalar1-major[1][4]*scalar2;
    major[2][5]=major[2][5]*scalar1-major[2][4]*scalar2;

    scalar2=minor[0][3];
    minor[1][3]=minor[1][3]*scalar1-minor[1][0]*scalar2;

    major[0][3]=major[0][3]*scalar1-major[0][0]*scalar2;
    major[1][3]=major[1][3]*scalar1-major[1][0]*scalar2;
    major[2][3]=major[2][3]*scalar1-major[2][0]*scalar2;

    major[0][7]=major[0][7]*scalar1-major[0][4]*scalar2;
    major[1][7]=major[1][7]*scalar1-major[1][4]*scalar2;
    major[2][7]=major[2][7]*scalar1-major[2][4]*scalar2;

    /**
     * Eliminate column 1 of rows 0 and 3
     * R(3)=y1'*R(3)-y3'*R(1)
     * R(0)=y1'*R(0)-(y0-y2)*R(1)
     */

    scalar1=minor[1][1];scalar2=minor[1][3];
    major[0][3]=major[0][3]*scalar1-major[0][1]*scalar2;
    major[1][3]=major[1][3]*scalar1-major[1][1]*scalar2;
    major[2][3]=major[2][3]*scalar1-major[2][1]*scalar2;

    major[0][7]=major[0][7]*scalar1-major[0][5]*scalar2;
    major[1][7]=major[1][7]*scalar1-major[1][5]*scalar2;
    major[2][7]=major[2][7]*scalar1-major[2][5]*scalar2;

    scalar2=minor[1][0];
    minor[0][0]=minor[0][0]*scalar1-minor[0][1]*scalar2;

    major[0][0]=major[0][0]*scalar1-major[0][1]*scalar2;
    major[1][0]=major[1][0]*scalar1-major[1][1]*scalar2;
    major[2][0]=major[2][0]*scalar1-major[2][1]*scalar2;

    major[0][4]=major[0][4]*scalar1-major[0][5]*scalar2;
    major[1][4]=major[1][4]*scalar1-major[1][5]*scalar2;
    major[2][4]=major[2][4]*scalar1-major[2][5]*scalar2;

    /**
     * Eliminate columns 0 and 1 of row 2
     * R(0)/=x0'
     * R(1)/=y1'
     * R(2)-= (x2*R(0) + y2*R(1))
     */

    scalar1=1.0f/minor[0][0];
    major[0][0]*=scalar1;
    major[1][0]*=scalar1;
    major[2][0]*=scalar1;
    major[0][4]*=scalar1;
    major[1][4]*=scalar1;
    major[2][4]*=scalar1;

    scalar1=1.0f/minor[1][1];
    major[0][1]*=scalar1;
    major[1][1]*=scalar1;
    major[2][1]*=scalar1;
    major[0][5]*=scalar1;
    major[1][5]*=scalar1;
    major[2][5]*=scalar1;


    scalar1=minor[0][2];scalar2=minor[1][2];
    major[0][2]-=major[0][0]*scalar1+major[0][1]*scalar2;
    major[1][2]-=major[1][0]*scalar1+major[1][1]*scalar2;
    major[2][2]-=major[2][0]*scalar1+major[2][1]*scalar2;

    major[0][6]-=major[0][4]*scalar1+major[0][5]*scalar2;
    major[1][6]-=major[1][4]*scalar1+major[1][5]*scalar2;
    major[2][6]-=major[2][4]*scalar1+major[2][5]*scalar2;

    /* Only major matters now. R(3) and R(7) correspond to the hollowed-out rows. */
    scalar1=major[0][7];
    major[1][7]/=scalar1;
    major[2][7]/=scalar1;

    scalar1=major[0][0];major[1][0]-=scalar1*major[1][7];major[2][0]-=scalar1*major[2][7];
    scalar1=major[0][1];major[1][1]-=scalar1*major[1][7];major[2][1]-=scalar1*major[2][7];
    scalar1=major[0][2];major[1][2]-=scalar1*major[1][7];major[2][2]-=scalar1*major[2][7];
    scalar1=major[0][3];major[1][3]-=scalar1*major[1][7];major[2][3]-=scalar1*major[2][7];
    scalar1=major[0][4];major[1][4]-=scalar1*major[1][7];major[2][4]-=scalar1*major[2][7];
    scalar1=major[0][5];major[1][5]-=scalar1*major[1][7];major[2][5]-=scalar1*major[2][7];
    scalar1=major[0][6];major[1][6]-=scalar1*major[1][7];major[2][6]-=scalar1*major[2][7];


    /* One column left (Two in fact, but the last one is the homography) */
    scalar1=major[1][3];

    major[2][3]/=scalar1;
    scalar1=major[1][0];major[2][0]-=scalar1*major[2][3];
    scalar1=major[1][1];major[2][1]-=scalar1*major[2][3];
    scalar1=major[1][2];major[2][2]-=scalar1*major[2][3];
    scalar1=major[1][4];major[2][4]-=scalar1*major[2][3];
    scalar1=major[1][5];major[2][5]-=scalar1*major[2][3];
    scalar1=major[1][6];major[2][6]-=scalar1*major[2][3];
    scalar1=major[1][7];major[2][7]-=scalar1*major[2][3];


    /* Homography is done. */
    H[0]=major[2][0];
    H[1]=major[2][1];
    H[2]=major[2][2];

    H[3]=major[2][4];
    H[4]=major[2][5];
    H[5]=major[2][6];

    H[6]=major[2][7];
    H[7]=major[2][3];
    H[8]=1.0;
}


/**
 * Returns whether refinement is possible.
 *
 * NB This is separate from whether it is *enabled*.
 *
 * @return 0 if refinement isn't possible, non-zero otherwise.
 *
 * Reads    (direct): best.numInl
 * Reads   (callees): None.
 * Writes   (direct): None.
 * Writes  (callees): None.
 */

inline int    RHO_HEST_REFC::canRefine(void){
    /**
     * If we only have 4 matches, GE's result is already optimal and cannot
     * be refined any further.
     */

    return best.numInl > (unsigned)SMPL_SIZE;
}


/**
 * Refines the best-so-far homography (p->best.H).
 *
 * Reads    (direct): best.H, arg.src, arg.dst, best.inl, arg.N, lm.JtJ,
 *                    lm.Jte, lm.tmp1
 * Reads   (callees): None.
 * Writes   (direct): best.H, lm.JtJ, lm.Jte, lm.tmp1
 * Writes  (callees): None.
 */

inline void   RHO_HEST_REFC::refine(void){
    int         i;
    float       S, newS;  /* Sum of squared errors */
    float       gain;     /* Gain-parameter. */
    float       L  = 100.0f;/* Lambda of LevMarq */
    float dH[8], newH[8];

    /**
     * Iteratively refine the homography.
     */
    /* Find initial conditions */
    sacCalcJacobianErrors(best.H, arg.src, arg.dst, best.inl, arg.N,
                          lm.JtJ, lm.Jte,  &S);

    /*Levenberg-Marquardt Loop.*/
    for(i=0;i<MAXLEVMARQITERS;i++){
        /**
         * Attempt a step given current state
         *   - Jacobian-x-Jacobian   (JtJ)
         *   - Jacobian-x-error      (Jte)
         *   - Sum of squared errors (S)
         * and current parameter
         *   - Lambda (L)
         * .
         *
         * This is done by solving the system of equations
         *     Ax = b
         * where A (JtJ) and b (Jte) are sourced from our current state, and
         * the solution x becomes a step (dH) that is applied to best.H in
         * order to compute a candidate homography (newH).
         *
         * The system above is solved by Cholesky decomposition of a
         * sufficently-damped JtJ into a lower-triangular matrix (and its
         * transpose), whose inverse is then computed. This inverse (and its
         * transpose) then multiply Jte in order to find dH.
         */

        while(!sacChol8x8Damped(lm.JtJ, L, lm.tmp1)){
            L *= 2.0f;
        }
        sacTRInv8x8   (lm.tmp1, lm.tmp1);
        sacTRISolve8x8(lm.tmp1, lm.Jte,  dH);
        sacSub8x1     (newH,       best.H,  dH);
        sacCalcJacobianErrors(newH, arg.src, arg.dst, best.inl, arg.N,
                              NULL, NULL, &newS);
        gain = sacLMGain(dH, lm.Jte, S, newS, L);
        /*printf("Lambda: %12.6f  S: %12.6f  newS: %12.6f  Gain: %12.6f\n",
                 L, S, newS, gain);*/

        /**
         * If the gain is positive (i.e., the new Sum of Square Errors (newS)
         * corresponding to newH is lower than the previous one (S) ), save
         * the current state and accept the new step dH.
         *
         * If the gain is below LM_GAIN_LO, damp more (increase L), even if the
         * gain was positive. If the gain is above LM_GAIN_HI, damp less
         * (decrease L). Otherwise the gain is left unchanged.
         */

        if(gain < LM_GAIN_LO){
            L *= 8;
            if(L>1000.0f/FLT_EPSILON){
                break;/* FIXME: Most naive termination criterion imaginable. */
            }
        }else if(gain > LM_GAIN_HI){
            L *= 0.5;
        }

        if(gain > 0){
            S = newS;
            memcpy(best.H, newH, sizeof(newH));
            sacCalcJacobianErrors(best.H, arg.src, arg.dst, best.inl, arg.N,
                                  lm.JtJ, lm.Jte,  &S);
        }
    }
}


/**
 * Compute directly the JtJ, Jte and sum-of-squared-error for a given
 * homography and set of inliers.
 *
 * This is possible because the product of J and its transpose as well as with
 * the error and the sum-of-squared-error can all be computed additively
 * (match-by-match), as one would intuitively expect; All matches make
 * contributions to the error independently of each other.
 *
 * What this allows is a constant-space implementation of Lev-Marq that can
 * nevertheless be vectorized if need be.
 */

static inline void   sacCalcJacobianErrors(const float* H,
                                           const float* src,
                                           const float* dst,
                                           const char*  inl,
                                           unsigned     N,
                                           float     (* JtJ)[8],
                                           float*       Jte,
                                           float*       Sp){
    unsigned i;
    float    S;

    /* Zero out JtJ, Jte and S */
    if(JtJ){memset(JtJ, 0, 8*8*sizeof(float));}
    if(Jte){memset(Jte, 0, 8*1*sizeof(float));}
    S = 0.0f;

    /* Additively compute JtJ and Jte */
    for(i=0;i<N;i++){
        /* Skip outliers */
        if(!inl[i]){
            continue;
        }

        /**
         * Otherwise, compute additively the upper triangular matrix JtJ and
         * the Jtd vector within the following formula:
         *
         *     LaTeX:
         *     (J^{T}J + \lambda \diag( J^{T}J )) \beta = J^{T}[ y - f(\Beta) ]
         *     Simplified ASCII:
         *     (JtJ + L*diag(JtJ)) beta = Jt e, where e (error) is y-f(Beta).
         *
         * For this we need to calculate
         *     1) The 2D error (e) of the homography on the current point i
         *        using the current parameters Beta.
         *     2) The derivatives (J) of the error on the current point i under
         *        perturbations of the current parameters Beta.
         * Accumulate products of the error times the derivative to Jte, and
         * products of the derivatives to JtJ.
         */

        /* Compute Squared Error */
        float x       = src[2*i+0];
        float y       = src[2*i+1];
        float X       = dst[2*i+0];
        float Y       = dst[2*i+1];
        float W       = (H[6]*x + H[7]*y + 1.0f);
        float iW      = fabs(W) > FLT_EPSILON ? 1.0f/W : 0;

        float reprojX = (H[0]*x + H[1]*y + H[2]) * iW;
        float reprojY = (H[3]*x + H[4]*y + H[5]) * iW;

        float eX      = reprojX - X;
        float eY      = reprojY - Y;
        float e       = eX*eX + eY*eY;
        S            += e;

        /* Compute Jacobian */
        if(JtJ || Jte){
            float dxh11 = x          * iW;
            float dxh12 = y          * iW;
            float dxh13 =              iW;
          /*float dxh21 = 0.0f;*/
          /*float dxh22 = 0.0f;*/
          /*float dxh23 = 0.0f;*/
            float dxh31 = -reprojX*x * iW;
            float dxh32 = -reprojX*y * iW;

          /*float dyh11 = 0.0f;*/
          /*float dyh12 = 0.0f;*/
          /*float dyh13 = 0.0f;*/
            float dyh21 = x          * iW;
            float dyh22 = y          * iW;
            float dyh23 =              iW;
            float dyh31 = -reprojY*x * iW;
            float dyh32 = -reprojY*y * iW;

            /* Update Jte:          X             Y   */
            if(Jte){
                Jte[0]    += eX   *dxh11              ;/*  +0 */
                Jte[1]    += eX   *dxh12              ;/*  +0 */
                Jte[2]    += eX   *dxh13              ;/*  +0 */
                Jte[3]    +=               eY   *dyh21;/* 0+  */
                Jte[4]    +=               eY   *dyh22;/* 0+  */
                Jte[5]    +=               eY   *dyh23;/* 0+  */
                Jte[6]    += eX   *dxh31 + eY   *dyh31;/*  +  */
                Jte[7]    += eX   *dxh32 + eY   *dyh32;/*  +  */
            }

            /* Update JtJ:          X             Y    */
            if(JtJ){
                JtJ[0][0] += dxh11*dxh11              ;/*  +0 */

                JtJ[1][0] += dxh11*dxh12              ;/*  +0 */
                JtJ[1][1] += dxh12*dxh12              ;/*  +0 */

                JtJ[2][0] += dxh11*dxh13              ;/*  +0 */
                JtJ[2][1] += dxh12*dxh13              ;/*  +0 */
                JtJ[2][2] += dxh13*dxh13              ;/*  +0 */

              /*JtJ[3][0] +=                          ;   0+0 */
              /*JtJ[3][1] +=                          ;   0+0 */
              /*JtJ[3][2] +=                          ;   0+0 */
                JtJ[3][3] +=               dyh21*dyh21;/* 0+  */

              /*JtJ[4][0] +=                          ;   0+0 */
              /*JtJ[4][1] +=                          ;   0+0 */
              /*JtJ[4][2] +=                          ;   0+0 */
                JtJ[4][3] +=               dyh21*dyh22;/* 0+  */
                JtJ[4][4] +=               dyh22*dyh22;/* 0+  */

              /*JtJ[5][0] +=                          ;   0+0 */
              /*JtJ[5][1] +=                          ;   0+0 */
              /*JtJ[5][2] +=                          ;   0+0 */
                JtJ[5][3] +=               dyh21*dyh23;/* 0+  */
                JtJ[5][4] +=               dyh22*dyh23;/* 0+  */
                JtJ[5][5] +=               dyh23*dyh23;/* 0+  */

                JtJ[6][0] += dxh11*dxh31              ;/*  +0 */
                JtJ[6][1] += dxh12*dxh31              ;/*  +0 */
                JtJ[6][2] += dxh13*dxh31              ;/*  +0 */
                JtJ[6][3] +=               dyh21*dyh31;/* 0+  */
                JtJ[6][4] +=               dyh22*dyh31;/* 0+  */
                JtJ[6][5] +=               dyh23*dyh31;/* 0+  */
                JtJ[6][6] += dxh31*dxh31 + dyh31*dyh31;/*  +  */

                JtJ[7][0] += dxh11*dxh32              ;/*  +0 */
                JtJ[7][1] += dxh12*dxh32              ;/*  +0 */
                JtJ[7][2] += dxh13*dxh32              ;/*  +0 */
                JtJ[7][3] +=               dyh21*dyh32;/* 0+  */
                JtJ[7][4] +=               dyh22*dyh32;/* 0+  */
                JtJ[7][5] +=               dyh23*dyh32;/* 0+  */
                JtJ[7][6] += dxh31*dxh32 + dyh31*dyh32;/*  +  */
                JtJ[7][7] += dxh32*dxh32 + dyh32*dyh32;/*  +  */
            }
        }
    }

    if(Sp){*Sp = S;}
}


/**
 * Compute the Levenberg-Marquardt "gain" obtained by the given step dH.
 *
 * Drawn from http://www2.imm.dtu.dk/documents/ftp/tr99/tr05_99.pdf.
 */

static inline float  sacLMGain(const float*  dH,
                               const float*  Jte,
                               const float   S,
                               const float   newS,
                               const float   lambda){
    float dS = S-newS;
    float dL = 0;
    int i;

    /* Compute h^t h... */
    for(i=0;i<8;i++){
        dL += dH[i]*dH[i];
    }
    /* Compute mu * h^t h... */
    dL *= lambda;
    /* Subtract h^t F'... */
    for(i=0;i<8;i++){
        dL += dH[i]*Jte[i];/* += as opposed to -=, since dH we compute is
                              opposite sign. */
    }
    /* Multiply by 1/2... */
    dL *= 0.5;

    /* Return gain as S-newS / L0 - LH. */
    return fabs(dL) < FLT_EPSILON ? dS : dS / dL;
}


/**
 * Cholesky decomposition on 8x8 real positive-definite matrix defined by its
 * lower-triangular half. Outputs L, the lower triangular part of the
 * decomposition.
 *
 * A and L can overlap fully (in-place) or not at all, but may not partially
 * overlap.
 *
 * For damping, the diagonal elements are scaled by 1.0 + lambda.
 *
 * Returns zero if decomposition unsuccessful, and non-zero otherwise.
 *
 * Source: http://en.wikipedia.org/wiki/Cholesky_decomposition#
 * The_Cholesky.E2.80.93Banachiewicz_and_Cholesky.E2.80.93Crout_algorithms
 */

static inline int    sacChol8x8Damped(const float (*A)[8],
                                      float         lambda,
                                      float       (*L)[8]){
    const int N = 8;
    int i, j, k;
    float  lambdap1 = lambda + 1.0f;
    float  x;

    for(i=0;i<N;i++){/* Row */
        /* Pre-diagonal elements */
        for(j=0;j<i;j++){
            x = A[i][j];               /* Aij */
            for(k=0;k<j;k++){
                x -= L[i][k] * L[j][k];/* - Sum_{k=0..j-1} Lik*Ljk */
            }
            L[i][j] = x / L[j][j];     /* Lij = ... / Ljj */
        }

        /* Diagonal element */
        {j = i;
            x = A[j][j] * lambdap1;    /* Ajj */
            for(k=0;k<j;k++){
                x -= L[j][k] * L[j][k];/* - Sum_{k=0..j-1} Ljk^2 */
            }
            if(x<0){
                return 0;
            }
            L[j][j] = sqrtf(x);        /* Ljj = sqrt( ... ) */
        }
    }

    return 1;
}


/**
 * Invert lower-triangular 8x8 matrix L into lower-triangular matrix M.
 *
 * L and M can overlap fully (in-place) or not at all, but may not partially
 * overlap.
 *
 * Uses formulation from
 * http://www.cs.berkeley.edu/~knight/knight_math221_poster.pdf
 * , adjusted for the fact that A^T^-1 = A^-1^T. Thus:
 *
 * U11    U12                   U11^-1   -U11^-1*U12*U22^-1
 *                ->
 *  0     U22                     0            U22^-1
 *
 * Becomes
 *
 * L11     0                    L11^-1           0
 *                ->
 * L21    L22            -L22^-1*L21*L11^-1    L22^-1
 *
 * Since
 *
 * ( -L11^T^-1*L21^T*L22^T^-1 )^T = -L22^T^-1^T*L21^T^T*L11^T^-1^T
 *                                = -L22^T^T^-1*L21^T^T*L11^T^T^-1
 *                                = -L22^-1*L21*L11^-1
 */

static inline void   sacTRInv8x8(const float (*L)[8],
                                 float       (*M)[8]){
    float s[2][2], t[2][2];
    float u[4][4], v[4][4];

    /*
        L00  0   0   0   0   0   0   0
        L10 L11  0   0   0   0   0   0
        L20 L21 L22  0   0   0   0   0
        L30 L31 L32 L33  0   0   0   0
        L40 L41 L42 L43 L44  0   0   0
        L50 L51 L52 L53 L54 L55  0   0
        L60 L61 L62 L63 L64 L65 L66  0
        L70 L71 L72 L73 L74 L75 L76 L77
    */

    /* Invert 4*2 1x1 matrices; Starts recursion. */
    M[0][0] = 1.0f/L[0][0];
    M[1][1] = 1.0f/L[1][1];
    M[2][2] = 1.0f/L[2][2];
    M[3][3] = 1.0f/L[3][3];
    M[4][4] = 1.0f/L[4][4];
    M[5][5] = 1.0f/L[5][5];
    M[6][6] = 1.0f/L[6][6];
    M[7][7] = 1.0f/L[7][7];

    /*
        M00  0   0   0   0   0   0   0
        L10 M11  0   0   0   0   0   0
        L20 L21 M22  0   0   0   0   0
        L30 L31 L32 M33  0   0   0   0
        L40 L41 L42 L43 M44  0   0   0
        L50 L51 L52 L53 L54 M55  0   0
        L60 L61 L62 L63 L64 L65 M66  0
        L70 L71 L72 L73 L74 L75 L76 M77
    */

    /* 4*2 Matrix products of 1x1 matrices */
    M[1][0] = -M[1][1]*L[1][0]*M[0][0];
    M[3][2] = -M[3][3]*L[3][2]*M[2][2];
    M[5][4] = -M[5][5]*L[5][4]*M[4][4];
    M[7][6] = -M[7][7]*L[7][6]*M[6][6];

    /*
        M00  0   0   0   0   0   0   0
        M10 M11  0   0   0   0   0   0
        L20 L21 M22  0   0   0   0   0
        L30 L31 M32 M33  0   0   0   0
        L40 L41 L42 L43 M44  0   0   0
        L50 L51 L52 L53 M54 M55  0   0
        L60 L61 L62 L63 L64 L65 M66  0
        L70 L71 L72 L73 L74 L75 M76 M77
    */

    /* 2*2 Matrix products of 2x2 matrices */

    /*
       (M22  0 )   (L20 L21)   (M00  0 )
     - (M32 M33) x (L30 L31) x (M10 M11)
    */

    s[0][0] = M[2][2]*L[2][0];
    s[0][1] = M[2][2]*L[2][1];
    s[1][0] = M[3][2]*L[2][0]+M[3][3]*L[3][0];
    s[1][1] = M[3][2]*L[2][1]+M[3][3]*L[3][1];

    t[0][0] = s[0][0]*M[0][0]+s[0][1]*M[1][0];
    t[0][1] =                 s[0][1]*M[1][1];
    t[1][0] = s[1][0]*M[0][0]+s[1][1]*M[1][0];
    t[1][1] =                 s[1][1]*M[1][1];

    M[2][0] = -t[0][0];
    M[2][1] = -t[0][1];
    M[3][0] = -t[1][0];
    M[3][1] = -t[1][1];

    /*
       (M66  0 )   (L64 L65)   (M44  0 )
     - (L76 M77) x (L74 L75) x (M54 M55)
    */

    s[0][0] = M[6][6]*L[6][4];
    s[0][1] = M[6][6]*L[6][5];
    s[1][0] = M[7][6]*L[6][4]+M[7][7]*L[7][4];
    s[1][1] = M[7][6]*L[6][5]+M[7][7]*L[7][5];

    t[0][0] = s[0][0]*M[4][4]+s[0][1]*M[5][4];
    t[0][1] =                 s[0][1]*M[5][5];
    t[1][0] = s[1][0]*M[4][4]+s[1][1]*M[5][4];
    t[1][1] =                 s[1][1]*M[5][5];

    M[6][4] = -t[0][0];
    M[6][5] = -t[0][1];
    M[7][4] = -t[1][0];
    M[7][5] = -t[1][1];

    /*
        M00  0   0   0   0   0   0   0
        M10 M11  0   0   0   0   0   0
        M20 M21 M22  0   0   0   0   0
        M30 M31 M32 M33  0   0   0   0
        L40 L41 L42 L43 M44  0   0   0
        L50 L51 L52 L53 M54 M55  0   0
        L60 L61 L62 L63 M64 M65 M66  0
        L70 L71 L72 L73 M74 M75 M76 M77
    */

    /* 1*2 Matrix products of 4x4 matrices */

    /*
       (M44  0   0   0 )   (L40 L41 L42 L43)   (M00  0   0   0 )
       (M54 M55  0   0 )   (L50 L51 L52 L53)   (M10 M11  0   0 )
       (M64 M65 M66  0 )   (L60 L61 L62 L63)   (M20 M21 M22  0 )
     - (M74 M75 M76 M77) x (L70 L71 L72 L73) x (M30 M31 M32 M33)
    */

    u[0][0] = M[4][4]*L[4][0];
    u[0][1] = M[4][4]*L[4][1];
    u[0][2] = M[4][4]*L[4][2];
    u[0][3] = M[4][4]*L[4][3];
    u[1][0] = M[5][4]*L[4][0]+M[5][5]*L[5][0];
    u[1][1] = M[5][4]*L[4][1]+M[5][5]*L[5][1];
    u[1][2] = M[5][4]*L[4][2]+M[5][5]*L[5][2];
    u[1][3] = M[5][4]*L[4][3]+M[5][5]*L[5][3];
    u[2][0] = M[6][4]*L[4][0]+M[6][5]*L[5][0]+M[6][6]*L[6][0];
    u[2][1] = M[6][4]*L[4][1]+M[6][5]*L[5][1]+M[6][6]*L[6][1];
    u[2][2] = M[6][4]*L[4][2]+M[6][5]*L[5][2]+M[6][6]*L[6][2];
    u[2][3] = M[6][4]*L[4][3]+M[6][5]*L[5][3]+M[6][6]*L[6][3];
    u[3][0] = M[7][4]*L[4][0]+M[7][5]*L[5][0]+M[7][6]*L[6][0]+M[7][7]*L[7][0];
    u[3][1] = M[7][4]*L[4][1]+M[7][5]*L[5][1]+M[7][6]*L[6][1]+M[7][7]*L[7][1];
    u[3][2] = M[7][4]*L[4][2]+M[7][5]*L[5][2]+M[7][6]*L[6][2]+M[7][7]*L[7][2];
    u[3][3] = M[7][4]*L[4][3]+M[7][5]*L[5][3]+M[7][6]*L[6][3]+M[7][7]*L[7][3];

    v[0][0] = u[0][0]*M[0][0]+u[0][1]*M[1][0]+u[0][2]*M[2][0]+u[0][3]*M[3][0];
    v[0][1] =                 u[0][1]*M[1][1]+u[0][2]*M[2][1]+u[0][3]*M[3][1];
    v[0][2] =                                 u[0][2]*M[2][2]+u[0][3]*M[3][2];
    v[0][3] =                                                 u[0][3]*M[3][3];
    v[1][0] = u[1][0]*M[0][0]+u[1][1]*M[1][0]+u[1][2]*M[2][0]+u[1][3]*M[3][0];
    v[1][1] =                 u[1][1]*M[1][1]+u[1][2]*M[2][1]+u[1][3]*M[3][1];
    v[1][2] =                                 u[1][2]*M[2][2]+u[1][3]*M[3][2];
    v[1][3] =                                                 u[1][3]*M[3][3];
    v[2][0] = u[2][0]*M[0][0]+u[2][1]*M[1][0]+u[2][2]*M[2][0]+u[2][3]*M[3][0];
    v[2][1] =                 u[2][1]*M[1][1]+u[2][2]*M[2][1]+u[2][3]*M[3][1];
    v[2][2] =                                 u[2][2]*M[2][2]+u[2][3]*M[3][2];
    v[2][3] =                                                 u[2][3]*M[3][3];
    v[3][0] = u[3][0]*M[0][0]+u[3][1]*M[1][0]+u[3][2]*M[2][0]+u[3][3]*M[3][0];
    v[3][1] =                 u[3][1]*M[1][1]+u[3][2]*M[2][1]+u[3][3]*M[3][1];
    v[3][2] =                                 u[3][2]*M[2][2]+u[3][3]*M[3][2];
    v[3][3] =                                                 u[3][3]*M[3][3];

    M[4][0] = -v[0][0];
    M[4][1] = -v[0][1];
    M[4][2] = -v[0][2];
    M[4][3] = -v[0][3];
    M[5][0] = -v[1][0];
    M[5][1] = -v[1][1];
    M[5][2] = -v[1][2];
    M[5][3] = -v[1][3];
    M[6][0] = -v[2][0];
    M[6][1] = -v[2][1];
    M[6][2] = -v[2][2];
    M[6][3] = -v[2][3];
    M[7][0] = -v[3][0];
    M[7][1] = -v[3][1];
    M[7][2] = -v[3][2];
    M[7][3] = -v[3][3];

    /*
        M00  0   0   0   0   0   0   0
        M10 M11  0   0   0   0   0   0
        M20 M21 M22  0   0   0   0   0
        M30 M31 M32 M33  0   0   0   0
        M40 M41 M42 M43 M44  0   0   0
        M50 M51 M52 M53 M54 M55  0   0
        M60 M61 M62 M63 M64 M65 M66  0
        M70 M71 M72 M73 M74 M75 M76 M77
    */
}


/**
 * Solves dH = inv(JtJ) Jte. The argument lower-triangular matrix is the
 * inverse of L as produced by the Cholesky decomposition LL^T of the matrix
 * JtJ; Thus the operation performed here is a left-multiplication of a vector
 * by two triangular matrices. The math is below:
 *
 * JtJ      = LL^T
 * Linv     = L^-1
 * (JtJ)^-1 = (LL^T)^-1
 *          = (L^T^-1)(Linv)
 *          = (Linv^T)(Linv)
 * dH       = ((JtJ)^-1) (Jte)
 *          = (Linv^T)(Linv) (Jte)
 *
 * where J is nx8, Jt is 8xn, JtJ is 8x8 PD, e is nx1, Jte is 8x1, L is lower
 * triangular 8x8 and dH is 8x1.
 */

static inline void   sacTRISolve8x8(const float (*L)[8],
                                    const float*  Jte,
                                    float*        dH){
    float t[8];

    t[0]  = L[0][0]*Jte[0];
    t[1]  = L[1][0]*Jte[0]+L[1][1]*Jte[1];
    t[2]  = L[2][0]*Jte[0]+L[2][1]*Jte[1]+L[2][2]*Jte[2];
    t[3]  = L[3][0]*Jte[0]+L[3][1]*Jte[1]+L[3][2]*Jte[2]+L[3][3]*Jte[3];
    t[4]  = L[4][0]*Jte[0]+L[4][1]*Jte[1]+L[4][2]*Jte[2]+L[4][3]*Jte[3]+L[4][4]*Jte[4];
    t[5]  = L[5][0]*Jte[0]+L[5][1]*Jte[1]+L[5][2]*Jte[2]+L[5][3]*Jte[3]+L[5][4]*Jte[4]+L[5][5]*Jte[5];
    t[6]  = L[6][0]*Jte[0]+L[6][1]*Jte[1]+L[6][2]*Jte[2]+L[6][3]*Jte[3]+L[6][4]*Jte[4]+L[6][5]*Jte[5]+L[6][6]*Jte[6];
    t[7]  = L[7][0]*Jte[0]+L[7][1]*Jte[1]+L[7][2]*Jte[2]+L[7][3]*Jte[3]+L[7][4]*Jte[4]+L[7][5]*Jte[5]+L[7][6]*Jte[6]+L[7][7]*Jte[7];


    dH[0] = L[0][0]*t[0]+L[1][0]*t[1]+L[2][0]*t[2]+L[3][0]*t[3]+L[4][0]*t[4]+L[5][0]*t[5]+L[6][0]*t[6]+L[7][0]*t[7];
    dH[1] =              L[1][1]*t[1]+L[2][1]*t[2]+L[3][1]*t[3]+L[4][1]*t[4]+L[5][1]*t[5]+L[6][1]*t[6]+L[7][1]*t[7];
    dH[2] =                           L[2][2]*t[2]+L[3][2]*t[3]+L[4][2]*t[4]+L[5][2]*t[5]+L[6][2]*t[6]+L[7][2]*t[7];
    dH[3] =                                        L[3][3]*t[3]+L[4][3]*t[4]+L[5][3]*t[5]+L[6][3]*t[6]+L[7][3]*t[7];
    dH[4] =                                                     L[4][4]*t[4]+L[5][4]*t[5]+L[6][4]*t[6]+L[7][4]*t[7];
    dH[5] =                                                                  L[5][5]*t[5]+L[6][5]*t[6]+L[7][5]*t[7];
    dH[6] =                                                                               L[6][6]*t[6]+L[7][6]*t[7];
    dH[7] =                                                                                            L[7][7]*t[7];
}


/**
 * Subtract dH from H.
 */

static inline void   sacSub8x1(float* Hout, const float* H, const float* dH){
    Hout[0] = H[0] - dH[0];
    Hout[1] = H[1] - dH[1];
    Hout[2] = H[2] - dH[2];
    Hout[3] = H[3] - dH[3];
    Hout[4] = H[4] - dH[4];
    Hout[5] = H[5] - dH[5];
    Hout[6] = H[6] - dH[6];
    Hout[7] = H[7] - dH[7];
}


/* End namespace cv */
}