opencv/modules/video/src/tvl1flow.cpp

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/*
//
// This implementation is based on Javier Sánchez Pérez <jsanchez@dis.ulpgc.es> implementation.
// Original BSD license:
//
// Copyright (c) 2011, Javier Sánchez Pérez, Enric Meinhardt Llopis
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice, this
//   list of conditions and the following disclaimer.
//
// * Redistributions 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.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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*/

#include "precomp.hpp"

using namespace std;
using namespace cv;

namespace {

class OpticalFlowDual_TVL1 : public DenseOpticalFlow
{
public:
    OpticalFlowDual_TVL1();

    void calc(InputArray I0, InputArray I1, InputOutputArray flow);
    void collectGarbage();

    AlgorithmInfo* info() const;

protected:
    double tau;
    double lambda;
    double theta;
    int nscales;
    int warps;
    double epsilon;
    int iterations;
    bool useInitialFlow;

private:
    void procOneScale(const Mat_<float>& I0, const Mat_<float>& I1, Mat_<float>& u1, Mat_<float>& u2);

    std::vector<Mat_<float> > I0s;
    std::vector<Mat_<float> > I1s;
    std::vector<Mat_<float> > u1s;
    std::vector<Mat_<float> > u2s;

    Mat_<float> I1x_buf;
    Mat_<float> I1y_buf;

    Mat_<float> flowMap1_buf;
    Mat_<float> flowMap2_buf;

    Mat_<float> I1w_buf;
    Mat_<float> I1wx_buf;
    Mat_<float> I1wy_buf;

    Mat_<float> grad_buf;
    Mat_<float> rho_c_buf;

    Mat_<float> v1_buf;
    Mat_<float> v2_buf;

    Mat_<float> p11_buf;
    Mat_<float> p12_buf;
    Mat_<float> p21_buf;
    Mat_<float> p22_buf;

    Mat_<float> div_p1_buf;
    Mat_<float> div_p2_buf;

    Mat_<float> u1x_buf;
    Mat_<float> u1y_buf;
    Mat_<float> u2x_buf;
    Mat_<float> u2y_buf;
};

OpticalFlowDual_TVL1::OpticalFlowDual_TVL1()
{
    tau            = 0.25;
    lambda         = 0.15;
    theta          = 0.3;
    nscales        = 5;
    warps          = 5;
    epsilon        = 0.01;
    iterations     = 300;
    useInitialFlow = false;
}

void OpticalFlowDual_TVL1::calc(InputArray _I0, InputArray _I1, InputOutputArray _flow)
{
    Mat I0 = _I0.getMat();
    Mat I1 = _I1.getMat();

    CV_Assert( I0.type() == CV_8UC1 || I0.type() == CV_32FC1 );
    CV_Assert( I0.size() == I1.size() );
    CV_Assert( I0.type() == I1.type() );
    CV_Assert( !useInitialFlow || (_flow.size() == I0.size() && _flow.type() == CV_32FC2) );
    CV_Assert( nscales > 0 );

    // allocate memory for the pyramid structure
    I0s.resize(nscales);
    I1s.resize(nscales);
    u1s.resize(nscales);
    u2s.resize(nscales);

    I0.convertTo(I0s[0], I0s[0].depth(), I0.depth() == CV_8U ? 1.0 : 255.0);
    I1.convertTo(I1s[0], I1s[0].depth(), I1.depth() == CV_8U ? 1.0 : 255.0);

    u1s[0].create(I0.size());
    u2s[0].create(I0.size());

    if (useInitialFlow)
    {
        Mat_<float> mv[] = {u1s[0], u2s[0]};
        split(_flow.getMat(), mv);
    }

    I1x_buf.create(I0.size());
    I1y_buf.create(I0.size());

    flowMap1_buf.create(I0.size());
    flowMap2_buf.create(I0.size());

    I1w_buf.create(I0.size());
    I1wx_buf.create(I0.size());
    I1wy_buf.create(I0.size());

    grad_buf.create(I0.size());
    rho_c_buf.create(I0.size());

    v1_buf.create(I0.size());
    v2_buf.create(I0.size());

    p11_buf.create(I0.size());
    p12_buf.create(I0.size());
    p21_buf.create(I0.size());
    p22_buf.create(I0.size());

    div_p1_buf.create(I0.size());
    div_p2_buf.create(I0.size());

    u1x_buf.create(I0.size());
    u1y_buf.create(I0.size());
    u2x_buf.create(I0.size());
    u2y_buf.create(I0.size());

    // create the scales
    for (int s = 1; s < nscales; ++s)
    {
        pyrDown(I0s[s - 1], I0s[s]);
        pyrDown(I1s[s - 1], I1s[s]);

        if (I0s[s].cols < 16 || I0s[s].rows < 16)
        {
            nscales = s;
            break;
        }

        if (useInitialFlow)
        {
            pyrDown(u1s[s - 1], u1s[s]);
            pyrDown(u2s[s - 1], u2s[s]);

            multiply(u1s[s], Scalar::all(0.5), u1s[s]);
            multiply(u2s[s], Scalar::all(0.5), u2s[s]);
        }
        else
        {
            u1s[s].create(I0s[s].size());
            u2s[s].create(I0s[s].size());
        }
    }

    if (!useInitialFlow)
    {
        u1s[nscales-1].setTo(Scalar::all(0));
        u2s[nscales-1].setTo(Scalar::all(0));
    }

    // pyramidal structure for computing the optical flow
    for (int s = nscales - 1; s >= 0; --s)
    {
        // compute the optical flow at the current scale
        procOneScale(I0s[s], I1s[s], u1s[s], u2s[s]);

        // if this was the last scale, finish now
        if (s == 0)
            break;

        // otherwise, upsample the optical flow

        // zoom the optical flow for the next finer scale
        resize(u1s[s], u1s[s - 1], I0s[s - 1].size());
        resize(u2s[s], u2s[s - 1], I0s[s - 1].size());

        // scale the optical flow with the appropriate zoom factor
        multiply(u1s[s - 1], Scalar::all(2), u1s[s - 1]);
        multiply(u2s[s - 1], Scalar::all(2), u2s[s - 1]);
    }

    Mat uxy[] = {u1s[0], u2s[0]};
    merge(uxy, 2, _flow);
}

////////////////////////////////////////////////////////////
// buildFlowMap

struct BuildFlowMapBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> u1;
    Mat_<float> u2;
    mutable Mat_<float> map1;
    mutable Mat_<float> map2;
};

void BuildFlowMapBody::operator() (const Range& range) const
{
    for (int y = range.start; y < range.end; ++y)
    {
        const float* u1Row = u1[y];
        const float* u2Row = u2[y];

        float* map1Row = map1[y];
        float* map2Row = map2[y];

        for (int x = 0; x < u1.cols; ++x)
        {
            map1Row[x] = x + u1Row[x];
            map2Row[x] = y + u2Row[x];
        }
    }
}

void buildFlowMap(const Mat_<float>& u1, const Mat_<float>& u2, Mat_<float>& map1, Mat_<float>& map2)
{
    CV_DbgAssert( u2.size() == u1.size() );
    CV_DbgAssert( map1.size() == u1.size() );
    CV_DbgAssert( map2.size() == u1.size() );

    BuildFlowMapBody body;

    body.u1 = u1;
    body.u2 = u2;
    body.map1 = map1;
    body.map2 = map2;

    parallel_for_(Range(0, u1.rows), body);
}

////////////////////////////////////////////////////////////
// centeredGradient

struct CenteredGradientBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> src;
    mutable Mat_<float> dx;
    mutable Mat_<float> dy;
};

void CenteredGradientBody::operator() (const Range& range) const
{
    const int last_col = src.cols - 1;

    for (int y = range.start; y < range.end; ++y)
    {
        const float* srcPrevRow = src[y - 1];
        const float* srcCurRow = src[y];
        const float* srcNextRow = src[y + 1];

        float* dxRow = dx[y];
        float* dyRow = dy[y];

        for (int x = 1; x < last_col; ++x)
        {
            dxRow[x] = 0.5f * (srcCurRow[x + 1] - srcCurRow[x - 1]);
            dyRow[x] = 0.5f * (srcNextRow[x] - srcPrevRow[x]);
        }
    }
}

void centeredGradient(const Mat_<float>& src, Mat_<float>& dx, Mat_<float>& dy)
{
    CV_DbgAssert( src.rows > 2 && src.cols > 2 );
    CV_DbgAssert( dx.size() == src.size() );
    CV_DbgAssert( dy.size() == src.size() );

    const int last_row = src.rows - 1;
    const int last_col = src.cols - 1;

    // compute the gradient on the center body of the image
    {
        CenteredGradientBody body;

        body.src = src;
        body.dx = dx;
        body.dy = dy;

        parallel_for_(Range(1, last_row), body);
    }

    // compute the gradient on the first and last rows
    for (int x = 1; x < last_col; ++x)
    {
        dx(0, x) = 0.5f * (src(0, x + 1) - src(0, x - 1));
        dy(0, x) = 0.5f * (src(1, x) - src(0, x));

        dx(last_row, x) = 0.5f * (src(last_row, x + 1) - src(last_row, x - 1));
        dy(last_row, x) = 0.5f * (src(last_row, x) - src(last_row - 1, x));
    }

    // compute the gradient on the first and last columns
    for (int y = 1; y < last_row; ++y)
    {
        dx(y, 0) = 0.5f * (src(y, 1) - src(y, 0));
        dy(y, 0) = 0.5f * (src(y + 1, 0) - src(y - 1, 0));

        dx(y, last_col) = 0.5f * (src(y, last_col) - src(y, last_col - 1));
        dy(y, last_col) = 0.5f * (src(y + 1, last_col) - src(y - 1, last_col));
    }

    // compute the gradient at the four corners
    dx(0, 0) = 0.5f * (src(0, 1) - src(0, 0));
    dy(0, 0) = 0.5f * (src(1, 0) - src(0, 0));

    dx(0, last_col) = 0.5f * (src(0, last_col) - src(0, last_col - 1));
    dy(0, last_col) = 0.5f * (src(1, last_col) - src(0, last_col));

    dx(last_row, 0) = 0.5f * (src(last_row, 1) - src(last_row, 0));
    dy(last_row, 0) = 0.5f * (src(last_row, 0) - src(last_row - 1, 0));

    dx(last_row, last_col) = 0.5f * (src(last_row, last_col) - src(last_row, last_col - 1));
    dy(last_row, last_col) = 0.5f * (src(last_row, last_col) - src(last_row - 1, last_col));
}

////////////////////////////////////////////////////////////
// forwardGradient

struct ForwardGradientBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> src;
    mutable Mat_<float> dx;
    mutable Mat_<float> dy;
};

void ForwardGradientBody::operator() (const Range& range) const
{
    const int last_col = src.cols - 1;

    for (int y = range.start; y < range.end; ++y)
    {
        const float* srcCurRow = src[y];
        const float* srcNextRow = src[y + 1];

        float* dxRow = dx[y];
        float* dyRow = dy[y];

        for (int x = 0; x < last_col; ++x)
        {
            dxRow[x] = srcCurRow[x + 1] - srcCurRow[x];
            dyRow[x] = srcNextRow[x] - srcCurRow[x];
        }
    }
}

void forwardGradient(const Mat_<float>& src, Mat_<float>& dx, Mat_<float>& dy)
{
    CV_DbgAssert( src.rows > 2 && src.cols > 2 );
    CV_DbgAssert( dx.size() == src.size() );
    CV_DbgAssert( dy.size() == src.size() );

    const int last_row = src.rows - 1;
    const int last_col = src.cols - 1;

    // compute the gradient on the central body of the image
    {
        ForwardGradientBody body;

        body.src = src;
        body.dx = dx;
        body.dy = dy;

        parallel_for_(Range(0, last_row), body);
    }

    // compute the gradient on the last row
    for (int x = 0; x < last_col; ++x)
    {
        dx(last_row, x) = src(last_row, x + 1) - src(last_row, x);
        dy(last_row, x) = 0.0f;
    }

    // compute the gradient on the last column
    for (int y = 0; y < last_row; ++y)
    {
        dx(y, last_col) = 0.0f;
        dy(y, last_col) = src(y + 1, last_col) - src(y, last_col);
    }

    dx(last_row, last_col) = 0.0f;
    dy(last_row, last_col) = 0.0f;
}

////////////////////////////////////////////////////////////
// divergence

struct DivergenceBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> v1;
    Mat_<float> v2;
    mutable Mat_<float> div;
};

void DivergenceBody::operator() (const Range& range) const
{
    for (int y = range.start; y < range.end; ++y)
    {
        const float* v1Row = v1[y];
        const float* v2PrevRow = v2[y - 1];
        const float* v2CurRow = v2[y];

        float* divRow = div[y];

        for(int x = 1; x < v1.cols; ++x)
        {
            const float v1x = v1Row[x] - v1Row[x - 1];
            const float v2y = v2CurRow[x] - v2PrevRow[x];

            divRow[x] = v1x + v2y;
        }
    }
}

void divergence(const Mat_<float>& v1, const Mat_<float>& v2, Mat_<float>& div)
{
    CV_DbgAssert( v1.rows > 2 && v1.cols > 2 );
    CV_DbgAssert( v2.size() == v1.size() );
    CV_DbgAssert( div.size() == v1.size() );

    {
        DivergenceBody body;

        body.v1 = v1;
        body.v2 = v2;
        body.div = div;

        parallel_for_(Range(1, v1.rows), body);
    }

    // compute the divergence on the first row
    for(int x = 1; x < v1.cols; ++x)
        div(0, x) = v1(0, x) - v1(0, x - 1) + v2(0, x);

    // compute the divergence on the first column
    for (int y = 1; y < v1.rows; ++y)
        div(y, 0) = v1(y, 0) + v2(y, 0) - v2(y - 1, 0);

    div(0, 0) = v1(0, 0) + v2(0, 0);
}

////////////////////////////////////////////////////////////
// calcGradRho

struct CalcGradRhoBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> I0;
    Mat_<float> I1w;
    Mat_<float> I1wx;
    Mat_<float> I1wy;
    Mat_<float> u1;
    Mat_<float> u2;
    mutable Mat_<float> grad;
    mutable Mat_<float> rho_c;
};

void CalcGradRhoBody::operator() (const Range& range) const
{
    for (int y = range.start; y < range.end; ++y)
    {
        const float* I0Row = I0[y];
        const float* I1wRow = I1w[y];
        const float* I1wxRow = I1wx[y];
        const float* I1wyRow = I1wy[y];
        const float* u1Row = u1[y];
        const float* u2Row = u2[y];

        float* gradRow = grad[y];
        float* rhoRow = rho_c[y];

        for (int x = 0; x < I0.cols; ++x)
        {
            const float Ix2 = I1wxRow[x] * I1wxRow[x];
            const float Iy2 = I1wyRow[x] * I1wyRow[x];

            // store the |Grad(I1)|^2
            gradRow[x] = Ix2 + Iy2;

            // compute the constant part of the rho function
            rhoRow[x] = (I1wRow[x] - I1wxRow[x] * u1Row[x] - I1wyRow[x] * u2Row[x] - I0Row[x]);
        }
    }
}

void calcGradRho(const Mat_<float>& I0, const Mat_<float>& I1w, const Mat_<float>& I1wx, const Mat_<float>& I1wy, const Mat_<float>& u1, const Mat_<float>& u2,
    Mat_<float>& grad, Mat_<float>& rho_c)
{
    CV_DbgAssert( I1w.size() == I0.size() );
    CV_DbgAssert( I1wx.size() == I0.size() );
    CV_DbgAssert( I1wy.size() == I0.size() );
    CV_DbgAssert( u1.size() == I0.size() );
    CV_DbgAssert( u2.size() == I0.size() );
    CV_DbgAssert( grad.size() == I0.size() );
    CV_DbgAssert( rho_c.size() == I0.size() );

    CalcGradRhoBody body;

    body.I0 = I0;
    body.I1w = I1w;
    body.I1wx = I1wx;
    body.I1wy = I1wy;
    body.u1 = u1;
    body.u2 = u2;
    body.grad = grad;
    body.rho_c = rho_c;

    parallel_for_(Range(0, I0.rows), body);
}

////////////////////////////////////////////////////////////
// estimateV

struct EstimateVBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> I1wx;
    Mat_<float> I1wy;
    Mat_<float> u1;
    Mat_<float> u2;
    Mat_<float> grad;
    Mat_<float> rho_c;
    mutable Mat_<float> v1;
    mutable Mat_<float> v2;
    float l_t;
};

void EstimateVBody::operator() (const Range& range) const
{
    for (int y = range.start; y < range.end; ++y)
    {
        const float* I1wxRow = I1wx[y];
        const float* I1wyRow = I1wy[y];
        const float* u1Row = u1[y];
        const float* u2Row = u2[y];
        const float* gradRow = grad[y];
        const float* rhoRow = rho_c[y];

        float* v1Row = v1[y];
        float* v2Row = v2[y];

        for (int x = 0; x < I1wx.cols; ++x)
        {
            const float rho = rhoRow[x] + (I1wxRow[x] * u1Row[x] + I1wyRow[x] * u2Row[x]);

            float d1 = 0.0f;
            float d2 = 0.0f;

            if (rho < -l_t * gradRow[x])
            {
                d1 = l_t * I1wxRow[x];
                d2 = l_t * I1wyRow[x];
            }
            else if (rho > l_t * gradRow[x])
            {
                d1 = -l_t * I1wxRow[x];
                d2 = -l_t * I1wyRow[x];
            }
            else if (gradRow[x] > numeric_limits<float>::epsilon())
            {
                float fi = -rho / gradRow[x];
                d1 = fi * I1wxRow[x];
                d2 = fi * I1wyRow[x];
            }

            v1Row[x] = u1Row[x] + d1;
            v2Row[x] = u2Row[x] + d2;
        }
    }
}

void estimateV(const Mat_<float>& I1wx, const Mat_<float>& I1wy, const Mat_<float>& u1, const Mat_<float>& u2, const Mat_<float>& grad, const Mat_<float>& rho_c,
               Mat_<float>& v1, Mat_<float>& v2, float l_t)
{
    CV_DbgAssert( I1wy.size() == I1wx.size() );
    CV_DbgAssert( u1.size() == I1wx.size() );
    CV_DbgAssert( u2.size() == I1wx.size() );
    CV_DbgAssert( grad.size() == I1wx.size() );
    CV_DbgAssert( rho_c.size() == I1wx.size() );
    CV_DbgAssert( v1.size() == I1wx.size() );
    CV_DbgAssert( v2.size() == I1wx.size() );

    EstimateVBody body;

    body.I1wx = I1wx;
    body.I1wy = I1wy;
    body.u1 = u1;
    body.u2 = u2;
    body.grad = grad;
    body.rho_c = rho_c;
    body.v1 = v1;
    body.v2 = v2;
    body.l_t = l_t;

    parallel_for_(Range(0, I1wx.rows), body);
}

////////////////////////////////////////////////////////////
// estimateU

float estimateU(const Mat_<float>& v1, const Mat_<float>& v2, const Mat_<float>& div_p1, const Mat_<float>& div_p2, Mat_<float>& u1, Mat_<float>& u2, float theta)
{
    CV_DbgAssert( v2.size() == v1.size() );
    CV_DbgAssert( div_p1.size() == v1.size() );
    CV_DbgAssert( div_p2.size() == v1.size() );
    CV_DbgAssert( u1.size() == v1.size() );
    CV_DbgAssert( u2.size() == v1.size() );

    float error = 0.0f;
    for (int y = 0; y < v1.rows; ++y)
    {
        const float* v1Row = v1[y];
        const float* v2Row = v2[y];
        const float* divP1Row = div_p1[y];
        const float* divP2Row = div_p2[y];

        float* u1Row = u1[y];
        float* u2Row = u2[y];

        for (int x = 0; x < v1.cols; ++x)
        {
            const float u1k = u1Row[x];
            const float u2k = u2Row[x];

            u1Row[x] = v1Row[x] + theta * divP1Row[x];
            u2Row[x] = v2Row[x] + theta * divP2Row[x];

            error += (u1Row[x] - u1k) * (u1Row[x] - u1k) + (u2Row[x] - u2k) * (u2Row[x] - u2k);
        }
    }

    return error;
}

////////////////////////////////////////////////////////////
// estimateDualVariables

struct EstimateDualVariablesBody : ParallelLoopBody
{
    void operator() (const Range& range) const;

    Mat_<float> u1x;
    Mat_<float> u1y;
    Mat_<float> u2x;
    Mat_<float> u2y;
    mutable Mat_<float> p11;
    mutable Mat_<float> p12;
    mutable Mat_<float> p21;
    mutable Mat_<float> p22;
    float taut;
};

void EstimateDualVariablesBody::operator() (const Range& range) const
{
    for (int y = range.start; y < range.end; ++y)
    {
        const float* u1xRow = u1x[y];
        const float* u1yRow = u1y[y];
        const float* u2xRow = u2x[y];
        const float* u2yRow = u2y[y];

        float* p11Row = p11[y];
        float* p12Row = p12[y];
        float* p21Row = p21[y];
        float* p22Row = p22[y];

        for (int x = 0; x < u1x.cols; ++x)
        {
            const float g1 = static_cast<float>(hypot(u1xRow[x], u1yRow[x]));
            const float g2 = static_cast<float>(hypot(u2xRow[x], u2yRow[x]));

            const float ng1  = 1.0f + taut * g1;
            const float ng2  = 1.0f + taut * g2;

            p11Row[x] = (p11Row[x] + taut * u1xRow[x]) / ng1;
            p12Row[x] = (p12Row[x] + taut * u1yRow[x]) / ng1;
            p21Row[x] = (p21Row[x] + taut * u2xRow[x]) / ng2;
            p22Row[x] = (p22Row[x] + taut * u2yRow[x]) / ng2;
        }
    }
}

void estimateDualVariables(const Mat_<float>& u1x, const Mat_<float>& u1y, const Mat_<float>& u2x, const Mat_<float>& u2y,
                           Mat_<float>& p11, Mat_<float>& p12, Mat_<float>& p21, Mat_<float>& p22, float taut)
{
    CV_DbgAssert( u1y.size() == u1x.size() );
    CV_DbgAssert( u2x.size() == u1x.size() );
    CV_DbgAssert( u2y.size() == u1x.size() );
    CV_DbgAssert( p11.size() == u1x.size() );
    CV_DbgAssert( p12.size() == u1x.size() );
    CV_DbgAssert( p21.size() == u1x.size() );
    CV_DbgAssert( p22.size() == u1x.size() );

    EstimateDualVariablesBody body;

    body.u1x = u1x;
    body.u1y = u1y;
    body.u2x = u2x;
    body.u2y = u2y;
    body.p11 = p11;
    body.p12 = p12;
    body.p21 = p21;
    body.p22 = p22;
    body.taut = taut;

    parallel_for_(Range(0, u1x.rows), body);
}

void OpticalFlowDual_TVL1::procOneScale(const Mat_<float>& I0, const Mat_<float>& I1, Mat_<float>& u1, Mat_<float>& u2)
{
    const float scaledEpsilon = static_cast<float>(epsilon * epsilon * I0.size().area());

    CV_DbgAssert( I1.size() == I0.size() );
    CV_DbgAssert( I1.type() == I0.type() );
    CV_DbgAssert( u1.size() == I0.size() );
    CV_DbgAssert( u2.size() == u1.size() );

    Mat_<float> I1x = I1x_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> I1y = I1y_buf(Rect(0, 0, I0.cols, I0.rows));
    centeredGradient(I1, I1x, I1y);

    Mat_<float> flowMap1 = flowMap1_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> flowMap2 = flowMap2_buf(Rect(0, 0, I0.cols, I0.rows));

    Mat_<float> I1w = I1w_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> I1wx = I1wx_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> I1wy = I1wy_buf(Rect(0, 0, I0.cols, I0.rows));

    Mat_<float> grad = grad_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> rho_c = rho_c_buf(Rect(0, 0, I0.cols, I0.rows));

    Mat_<float> v1 = v1_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> v2 = v2_buf(Rect(0, 0, I0.cols, I0.rows));

    Mat_<float> p11 = p11_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> p12 = p12_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> p21 = p21_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> p22 = p22_buf(Rect(0, 0, I0.cols, I0.rows));
    p11.setTo(Scalar::all(0));
    p12.setTo(Scalar::all(0));
    p21.setTo(Scalar::all(0));
    p22.setTo(Scalar::all(0));

    Mat_<float> div_p1 = div_p1_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> div_p2 = div_p2_buf(Rect(0, 0, I0.cols, I0.rows));

    Mat_<float> u1x = u1x_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> u1y = u1y_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> u2x = u2x_buf(Rect(0, 0, I0.cols, I0.rows));
    Mat_<float> u2y = u2y_buf(Rect(0, 0, I0.cols, I0.rows));

    const float l_t = static_cast<float>(lambda * theta);
    const float taut = static_cast<float>(tau / theta);

    for (int warpings = 0; warpings < warps; ++warpings)
    {
        // compute the warping of the target image and its derivatives
        buildFlowMap(u1, u2, flowMap1, flowMap2);
        remap(I1, I1w, flowMap1, flowMap2, INTER_CUBIC);
        remap(I1x, I1wx, flowMap1, flowMap2, INTER_CUBIC);
        remap(I1y, I1wy, flowMap1, flowMap2, INTER_CUBIC);

        calcGradRho(I0, I1w, I1wx, I1wy, u1, u2, grad, rho_c);

        float error = numeric_limits<float>::max();
        for (int n = 0; error > scaledEpsilon && n < iterations; ++n)
        {
            // estimate the values of the variable (v1, v2) (thresholding operator TH)
            estimateV(I1wx, I1wy, u1, u2, grad, rho_c, v1, v2, l_t);

            // compute the divergence of the dual variable (p1, p2)
            divergence(p11, p12, div_p1);
            divergence(p21, p22, div_p2);

            // estimate the values of the optical flow (u1, u2)
            error = estimateU(v1, v2, div_p1, div_p2, u1, u2, static_cast<float>(theta));

            // compute the gradient of the optical flow (Du1, Du2)
            forwardGradient(u1, u1x, u1y);
            forwardGradient(u2, u2x, u2y);

            // estimate the values of the dual variable (p1, p2)
            estimateDualVariables(u1x, u1y, u2x, u2y, p11, p12, p21, p22, taut);
        }
    }
}

void OpticalFlowDual_TVL1::collectGarbage()
{
    I0s.clear();
    I1s.clear();
    u1s.clear();
    u2s.clear();

    I1x_buf.release();
    I1y_buf.release();

    flowMap1_buf.release();
    flowMap2_buf.release();

    I1w_buf.release();
    I1wx_buf.release();
    I1wy_buf.release();

    grad_buf.release();
    rho_c_buf.release();

    v1_buf.release();
    v2_buf.release();

    p11_buf.release();
    p12_buf.release();
    p21_buf.release();
    p22_buf.release();

    div_p1_buf.release();
    div_p2_buf.release();

    u1x_buf.release();
    u1y_buf.release();
    u2x_buf.release();
    u2y_buf.release();
}

CV_INIT_ALGORITHM(OpticalFlowDual_TVL1, "DenseOpticalFlow.DualTVL1",
                  obj.info()->addParam(obj, "tau", obj.tau, false, 0, 0,
                                       "Time step of the numerical scheme");
                  obj.info()->addParam(obj, "lambda", obj.lambda, false, 0, 0,
                                       "Weight parameter for the data term, attachment parameter");
                  obj.info()->addParam(obj, "theta", obj.theta, false, 0, 0,
                                       "Weight parameter for (u - v)^2, tightness parameter");
                  obj.info()->addParam(obj, "nscales", obj.nscales, false, 0, 0,
                                       "Number of scales used to create the pyramid of images");
                  obj.info()->addParam(obj, "warps", obj.warps, false, 0, 0,
                                       "Number of warpings per scale");
                  obj.info()->addParam(obj, "epsilon", obj.epsilon, false, 0, 0,
                                       "Stopping criterion threshold used in the numerical scheme, which is a trade-off between precision and running time");
                  obj.info()->addParam(obj, "iterations", obj.iterations, false, 0, 0,
                                       "Stopping criterion iterations number used in the numerical scheme");
                  obj.info()->addParam(obj, "useInitialFlow", obj.useInitialFlow));

} // namespace

Ptr<DenseOpticalFlow> cv::createOptFlow_DualTVL1()
{
    return new OpticalFlowDual_TVL1;
}