527 lines
19 KiB
C++
527 lines
19 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file testSimilarity3.cpp
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* @brief Unit tests for Similarity3 class
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* @author Paul Drews
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* @author Zhaoyang Lv
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*/
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#include <gtsam_unstable/geometry/Similarity3.h>
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#include <gtsam/slam/BetweenFactor.h>
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#include <gtsam/nonlinear/NonlinearFactorGraph.h>
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#include <gtsam/nonlinear/ExpressionFactorGraph.h>
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#include <gtsam/nonlinear/Values.h>
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#include <gtsam/nonlinear/LevenbergMarquardtOptimizer.h>
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#include <gtsam/geometry/Pose3.h>
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#include <gtsam/inference/Symbol.h>
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#include <gtsam/base/numericalDerivative.h>
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#include <gtsam/base/testLie.h>
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#include <gtsam/base/Testable.h>
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#include <CppUnitLite/TestHarness.h>
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#include <boost/function.hpp>
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#include <boost/bind.hpp>
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using namespace gtsam;
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using namespace std;
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using symbol_shorthand::X;
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GTSAM_CONCEPT_TESTABLE_INST(Similarity3)
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static const Point3 P(0.2, 0.7, -2);
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static const Rot3 R = Rot3::Rodrigues(0.3, 0, 0);
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static const double s = 4;
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static const Similarity3 id;
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static const Similarity3 T1(R, Point3(3.5, -8.2, 4.2), 1);
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static const Similarity3 T2(Rot3::Rodrigues(0.3, 0.2, 0.1),
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Point3(3.5, -8.2, 4.2), 1);
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static const Similarity3 T3(Rot3::Rodrigues(-90, 0, 0), Point3(1, 2, 3), 1);
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static const Similarity3 T4(R, P, s);
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static const Similarity3 T5(R, P, 10);
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static const Similarity3 T6(Rot3(), Point3(1, 1, 0), 2); // Simpler transform
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const double degree = M_PI / 180;
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//******************************************************************************
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TEST(Similarity3, Concepts) {
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BOOST_CONCEPT_ASSERT((IsGroup<Similarity3 >));
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BOOST_CONCEPT_ASSERT((IsManifold<Similarity3 >));
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BOOST_CONCEPT_ASSERT((IsLieGroup<Similarity3 >));
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}
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//******************************************************************************
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TEST(Similarity3, Constructors) {
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Similarity3 sim3_Construct1;
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Similarity3 sim3_Construct2(s);
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Similarity3 sim3_Construct3(R, P, s);
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Similarity3 sim4_Construct4(R.matrix(), P, s);
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}
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//******************************************************************************
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TEST(Similarity3, Getters) {
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Similarity3 sim3_default;
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EXPECT(assert_equal(Rot3(), sim3_default.rotation()));
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EXPECT(assert_equal(Point3(0,0,0), sim3_default.translation()));
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EXPECT_DOUBLES_EQUAL(1.0, sim3_default.scale(), 1e-9);
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Similarity3 sim3(Rot3::Ypr(1, 2, 3), Point3(4, 5, 6), 7);
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EXPECT(assert_equal(Rot3::Ypr(1, 2, 3), sim3.rotation()));
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EXPECT(assert_equal(Point3(4, 5, 6), sim3.translation()));
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EXPECT_DOUBLES_EQUAL(7.0, sim3.scale(), 1e-9);
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}
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//******************************************************************************
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TEST(Similarity3, AdjointMap) {
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const Matrix4 T = T2.matrix();
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// Check Ad with actual definition
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Vector7 delta;
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delta << 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7;
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Matrix4 W = Similarity3::wedge(delta);
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Matrix4 TW = Similarity3::wedge(T2.AdjointMap() * delta);
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EXPECT(assert_equal(TW, Matrix4(T * W * T.inverse()), 1e-9));
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}
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//******************************************************************************
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TEST(Similarity3, inverse) {
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Similarity3 sim3(Rot3::Ypr(1, 2, 3).inverse(), Point3(4, 5, 6), 7);
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Matrix3 Re; // some values from matlab
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Re << -0.2248, 0.9024, -0.3676, -0.3502, -0.4269, -0.8337, -0.9093, -0.0587, 0.4120;
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Vector3 te(-9.8472, 59.7640, 10.2125);
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Similarity3 expected(Re, te, 1.0 / 7.0);
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EXPECT(assert_equal(expected, sim3.inverse(), 1e-4));
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EXPECT(assert_equal(sim3, sim3.inverse().inverse(), 1e-8));
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// test lie group inverse
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Matrix H1, H2;
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EXPECT(assert_equal(expected, sim3.inverse(H1), 1e-4));
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EXPECT(assert_equal(sim3, sim3.inverse().inverse(H2), 1e-8));
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}
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//******************************************************************************
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TEST(Similarity3, Multiplication) {
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Similarity3 test1(Rot3::Ypr(1, 2, 3).inverse(), Point3(4, 5, 6), 7);
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Similarity3 test2(Rot3::Ypr(1, 2, 3).inverse(), Point3(8, 9, 10), 11);
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Matrix3 re;
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re << 0.0688, 0.9863, -0.1496, -0.5665, -0.0848, -0.8197, -0.8211, 0.1412, 0.5530;
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Vector3 te(-13.6797, 3.2441, -5.7794);
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Similarity3 expected(re, te, 77);
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EXPECT(assert_equal(expected, test1 * test2, 1e-2));
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}
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//******************************************************************************
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TEST(Similarity3, Manifold) {
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EXPECT_LONGS_EQUAL(7, Similarity3::Dim());
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Vector z = Vector7::Zero();
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Similarity3 sim;
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EXPECT(sim.retract(z) == sim);
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Vector7 v = Vector7::Zero();
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v(6) = 2;
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Similarity3 sim2;
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EXPECT(sim2.retract(z) == sim2);
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EXPECT(assert_equal(z, sim2.localCoordinates(sim)));
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Similarity3 sim3 = Similarity3(Rot3(), Point3(1, 2, 3), 1);
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Vector v3(7);
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v3 << 0, 0, 0, 1, 2, 3, 0;
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EXPECT(assert_equal(v3, sim2.localCoordinates(sim3)));
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Similarity3 other = Similarity3(Rot3::Ypr(0.1, 0.2, 0.3), Point3(4, 5, 6), 1);
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Vector vlocal = sim.localCoordinates(other);
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EXPECT(assert_equal(sim.retract(vlocal), other, 1e-2));
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Similarity3 other2 = Similarity3(Rot3::Ypr(0.3, 0, 0), Point3(4, 5, 6), 1);
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Rot3 R = Rot3::Rodrigues(0.3, 0, 0);
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Vector vlocal2 = sim.localCoordinates(other2);
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EXPECT(assert_equal(sim.retract(vlocal2), other2, 1e-2));
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// TODO add unit tests for retract and localCoordinates
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}
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//******************************************************************************
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TEST( Similarity3, retract_first_order) {
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Similarity3 id;
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Vector v = Z_7x1;
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v(0) = 0.3;
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EXPECT(assert_equal(Similarity3(R, Point3(0,0,0), 1), id.retract(v), 1e-2));
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// v(3) = 0.2;
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// v(4) = 0.7;
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// v(5) = -2;
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// EXPECT(assert_equal(Similarity3(R, P, 1), id.retract(v), 1e-2));
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}
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//******************************************************************************
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TEST(Similarity3, localCoordinates_first_order) {
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Vector7 d12 = Vector7::Constant(0.1);
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d12(6) = 1.0;
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Similarity3 t1 = T1, t2 = t1.retract(d12);
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EXPECT(assert_equal(d12, t1.localCoordinates(t2)));
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}
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//******************************************************************************
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TEST(Similarity3, manifold_first_order) {
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Similarity3 t1 = T1;
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Similarity3 t2 = T3;
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Similarity3 origin;
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Vector d12 = t1.localCoordinates(t2);
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EXPECT(assert_equal(t2, t1.retract(d12)));
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Vector d21 = t2.localCoordinates(t1);
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EXPECT(assert_equal(t1, t2.retract(d21)));
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}
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//******************************************************************************
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// Return as a 4*4 Matrix
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TEST(Similarity3, Matrix) {
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Matrix4 expected;
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expected << 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0.5;
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Matrix4 actual = T6.matrix();
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EXPECT(assert_equal(expected, actual));
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}
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//*****************************************************************************
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// Exponential and log maps
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TEST(Similarity3, ExpLogMap) {
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Vector7 delta;
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delta << 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7;
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Vector7 actual = Similarity3::Logmap(Similarity3::Expmap(delta));
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EXPECT(assert_equal(delta, actual));
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Vector7 zeros;
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zeros << 0, 0, 0, 0, 0, 0, 0;
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Vector7 logIdentity = Similarity3::Logmap(Similarity3::identity());
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EXPECT(assert_equal(zeros, logIdentity));
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Similarity3 expZero = Similarity3::Expmap(zeros);
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Similarity3 ident = Similarity3::identity();
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EXPECT(assert_equal(expZero, ident));
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// Compare to matrix exponential, using expm in Lie.h
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EXPECT(
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assert_equal(expm<Similarity3>(delta), Similarity3::Expmap(delta), 1e-3));
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}
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//******************************************************************************
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// Group action on Point3 (with simpler transform)
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TEST(Similarity3, GroupAction) {
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EXPECT(assert_equal(Point3(2, 2, 0), T6 * Point3(0, 0, 0)));
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EXPECT(assert_equal(Point3(4, 2, 0), T6 * Point3(1, 0, 0)));
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// Test group action on R^4 via matrix representation
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Vector4 qh;
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qh << 1, 0, 0, 1;
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Vector4 ph;
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ph << 2, 1, 0, 0.5; // equivalent to Point3(4, 2, 0)
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EXPECT(assert_equal((Vector )ph, T6.matrix() * qh));
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// Test some more...
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Point3 pa = Point3(1, 0, 0);
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Similarity3 Ta(Rot3(), Point3(1, 2, 3), 1.0);
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Similarity3 Tb(Rot3(), Point3(1, 2, 3), 2.0);
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EXPECT(assert_equal(Point3(2, 2, 3), Ta.transformFrom(pa)));
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EXPECT(assert_equal(Point3(4, 4, 6), Tb.transformFrom(pa)));
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Similarity3 Tc(Rot3::Rz(M_PI / 2.0), Point3(1, 2, 3), 1.0);
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Similarity3 Td(Rot3::Rz(M_PI / 2.0), Point3(1, 2, 3), 2.0);
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EXPECT(assert_equal(Point3(1, 3, 3), Tc.transformFrom(pa)));
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EXPECT(assert_equal(Point3(2, 6, 6), Td.transformFrom(pa)));
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// Test derivative
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boost::function<Point3(Similarity3, Point3)> f = boost::bind(
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&Similarity3::transformFrom, _1, _2, boost::none, boost::none);
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Point3 q(1, 2, 3);
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for (const auto& T : { T1, T2, T3, T4, T5, T6 }) {
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Point3 q(1, 0, 0);
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Matrix H1 = numericalDerivative21<Point3, Similarity3, Point3>(f, T, q);
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Matrix H2 = numericalDerivative22<Point3, Similarity3, Point3>(f, T, q);
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Matrix actualH1, actualH2;
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T.transformFrom(q, actualH1, actualH2);
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EXPECT(assert_equal(H1, actualH1));
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EXPECT(assert_equal(H2, actualH2));
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}
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}
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//******************************************************************************
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// Group action on Pose3
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TEST(Similarity3, GroupActionPose3) {
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Similarity3 bSa(Rot3::Ry(180 * degree), Point3(2, 3, 5), 2.0);
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// Create source poses
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Pose3 Ta1(Rot3(), Point3(0, 0, 0));
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Pose3 Ta2(Rot3(-1, 0, 0, 0, -1, 0, 0, 0, 1), Point3(4, 0, 0));
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// Create destination poses
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Pose3 expected_Tb1(Rot3(-1, 0, 0, 0, 1, 0, 0, 0, -1), Point3(4, 6, 10));
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Pose3 expected_Tb2(Rot3(1, 0, 0, 0, -1, 0, 0, 0, -1), Point3(-4, 6, 10));
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EXPECT(assert_equal(expected_Tb1, bSa.transformFrom(Ta1)));
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EXPECT(assert_equal(expected_Tb2, bSa.transformFrom(Ta2)));
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}
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// Test left group action compatibility.
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// cSa*Ta = cSb*bSa*Ta
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TEST(Similarity3, GroupActionPose3_Compatibility) {
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Similarity3 bSa(Rot3::Ry(180 * degree), Point3(2, 3, 5), 2.0);
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Similarity3 cSb(Rot3::Ry(90 * degree), Point3(-10, -4, 0), 3.0);
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Similarity3 cSa(Rot3::Ry(270 * degree), Point3(0, 1, -2), 6.0);
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// Create poses
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Pose3 Ta1(Rot3(), Point3(0, 0, 0));
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Pose3 Ta2(Rot3(-1, 0, 0, 0, -1, 0, 0, 0, 1), Point3(4, 0, 0));
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Pose3 Tb1(Rot3(-1, 0, 0, 0, 1, 0, 0, 0, -1), Point3(4, 6, 10));
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Pose3 Tb2(Rot3(1, 0, 0, 0, -1, 0, 0, 0, -1), Point3(-4, 6, 10));
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Pose3 Tc1(Rot3(0, 0, -1, 0, 1, 0, 1, 0, 0), Point3(0, 6, -12));
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Pose3 Tc2(Rot3(0, 0, -1, 0, -1, 0, -1, 0, 0), Point3(0, 6, 12));
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EXPECT(assert_equal(Tc1, cSb.transformFrom(Tb1)));
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EXPECT(assert_equal(Tc2, cSb.transformFrom(Tb2)));
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EXPECT(assert_equal(cSa.transformFrom(Ta1), cSb.transformFrom(Tb1)));
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EXPECT(assert_equal(cSa.transformFrom(Ta2), cSb.transformFrom(Tb2)));
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}
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//******************************************************************************
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// Align with Point3 Pairs
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TEST(Similarity3, AlignPoint3_1) {
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Similarity3 expected_aSb(Rot3::Rz(-90 * degree), Point3(3, 4, 5), 2.0);
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Point3 b1(0, 0, 0), b2(3, 0, 0), b3(3, 0, 4);
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Point3Pair ab1(make_pair(expected_aSb.transformFrom(b1), b1));
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Point3Pair ab2(make_pair(expected_aSb.transformFrom(b2), b2));
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Point3Pair ab3(make_pair(expected_aSb.transformFrom(b3), b3));
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vector<Point3Pair> correspondences{ab1, ab2, ab3};
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Similarity3 actual_aSb = Similarity3::Align(correspondences);
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EXPECT(assert_equal(expected_aSb, actual_aSb));
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}
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TEST(Similarity3, AlignPoint3_2) {
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Similarity3 expected_aSb(Rot3(), Point3(10, 10, 0), 1.0);
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Point3 b1(0, 0, 0), b2(20, 10, 0), b3(10, 20, 0);
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Point3Pair ab1(make_pair(expected_aSb.transformFrom(b1), b1));
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Point3Pair ab2(make_pair(expected_aSb.transformFrom(b2), b2));
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Point3Pair ab3(make_pair(expected_aSb.transformFrom(b3), b3));
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vector<Point3Pair> correspondences{ab1, ab2, ab3};
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Similarity3 actual_aSb = Similarity3::Align(correspondences);
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EXPECT(assert_equal(expected_aSb, actual_aSb));
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}
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TEST(Similarity3, AlignPoint3_3) {
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Similarity3 expected_aSb(Rot3::RzRyRx(0.3, 0.2, 0.1), Point3(20, 10, 5), 1.0);
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Point3 b1(0, 0, 1), b2(10, 0, 2), b3(20, -10, 30);
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Point3Pair ab1(make_pair(expected_aSb.transformFrom(b1), b1));
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Point3Pair ab2(make_pair(expected_aSb.transformFrom(b2), b2));
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Point3Pair ab3(make_pair(expected_aSb.transformFrom(b3), b3));
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vector<Point3Pair> correspondences{ab1, ab2, ab3};
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Similarity3 actual_aSb = Similarity3::Align(correspondences);
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EXPECT(assert_equal(expected_aSb, actual_aSb));
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}
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//******************************************************************************
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// Align with Pose3 Pairs
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TEST(Similarity3, AlignPose3) {
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Similarity3 expected_aSb(Rot3::Ry(180 * degree), Point3(2, 3, 5), 2.0);
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// Create source poses
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Pose3 Ta1(Rot3(), Point3(0, 0, 0));
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Pose3 Ta2(Rot3(-1, 0, 0, 0, -1, 0, 0, 0, 1), Point3(4, 0, 0));
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// Create destination poses
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Pose3 Tb1(Rot3(-1, 0, 0, 0, 1, 0, 0, 0, -1), Point3(4, 6, 10));
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Pose3 Tb2(Rot3(1, 0, 0, 0, -1, 0, 0, 0, -1), Point3(-4, 6, 10));
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Pose3Pair bTa1(make_pair(Tb1, Ta1));
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Pose3Pair bTa2(make_pair(Tb2, Ta2));
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vector<Pose3Pair> correspondences{bTa1, bTa2};
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// Cayley transform cannot accommodate 180 degree rotations,
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// hence we only test for Expmap
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#ifdef GTSAM_ROT3_EXPMAP
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Similarity3 actual_aSb = Similarity3::Align(correspondences);
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EXPECT(assert_equal(expected_aSb, actual_aSb));
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#endif
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}
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//******************************************************************************
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// Test very simple prior optimization example
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TEST(Similarity3, Optimization) {
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// Create a PriorFactor with a Sim3 prior
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Similarity3 prior = Similarity3(Rot3::Ypr(0.1, 0.2, 0.3), Point3(1, 2, 3), 4);
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noiseModel::Isotropic::shared_ptr model = noiseModel::Isotropic::Sigma(7, 1);
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Symbol key('x', 1);
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// Create graph
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NonlinearFactorGraph graph;
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graph.addPrior(key, prior, model);
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// Create initial estimate with identity transform
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Values initial;
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initial.insert<Similarity3>(key, Similarity3());
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// Optimize
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Values result;
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LevenbergMarquardtParams params;
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params.setVerbosityLM("TRYCONFIG");
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result = LevenbergMarquardtOptimizer(graph, initial).optimize();
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// After optimization, result should be prior
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EXPECT(assert_equal(prior, result.at<Similarity3>(key), 1e-4));
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}
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//******************************************************************************
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// Test optimization with both Prior and BetweenFactors
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TEST(Similarity3, Optimization2) {
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Similarity3 prior = Similarity3();
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Similarity3 m1 = Similarity3(Rot3::Ypr(M_PI / 4.0, 0, 0), Point3(2.0, 0, 0),
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1.0);
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Similarity3 m2 = Similarity3(Rot3::Ypr(M_PI / 2.0, 0, 0),
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Point3(sqrt(8) * 0.9, 0, 0), 1.0);
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Similarity3 m3 = Similarity3(Rot3::Ypr(3 * M_PI / 4.0, 0, 0),
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Point3(sqrt(32) * 0.8, 0, 0), 1.0);
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Similarity3 m4 = Similarity3(Rot3::Ypr(M_PI / 2.0, 0, 0),
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Point3(6 * 0.7, 0, 0), 1.0);
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Similarity3 loop = Similarity3(1.42);
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//prior.print("Goal Transform");
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noiseModel::Isotropic::shared_ptr model = noiseModel::Isotropic::Sigma(7,
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0.01);
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SharedDiagonal betweenNoise = noiseModel::Diagonal::Sigmas(
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(Vector(7) << 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 10).finished());
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SharedDiagonal betweenNoise2 = noiseModel::Diagonal::Sigmas(
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(Vector(7) << 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0).finished());
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BetweenFactor<Similarity3> b1(X(1), X(2), m1, betweenNoise);
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BetweenFactor<Similarity3> b2(X(2), X(3), m2, betweenNoise);
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BetweenFactor<Similarity3> b3(X(3), X(4), m3, betweenNoise);
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BetweenFactor<Similarity3> b4(X(4), X(5), m4, betweenNoise);
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BetweenFactor<Similarity3> lc(X(5), X(1), loop, betweenNoise2);
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|
|
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// Create graph
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NonlinearFactorGraph graph;
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graph.addPrior(X(1), prior, model); // Prior !
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graph.push_back(b1);
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graph.push_back(b2);
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graph.push_back(b3);
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graph.push_back(b4);
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graph.push_back(lc);
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|
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//graph.print("Full Graph\n");
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Values initial;
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initial.insert<Similarity3>(X(1), Similarity3());
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initial.insert<Similarity3>(X(2),
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|
Similarity3(Rot3::Ypr(M_PI / 2.0, 0, 0), Point3(1, 0, 0), 1.1));
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initial.insert<Similarity3>(X(3),
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|
Similarity3(Rot3::Ypr(2.0 * M_PI / 2.0, 0, 0), Point3(0.9, 1.1, 0), 1.2));
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|
initial.insert<Similarity3>(X(4),
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|
Similarity3(Rot3::Ypr(3.0 * M_PI / 2.0, 0, 0), Point3(0, 1, 0), 1.3));
|
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initial.insert<Similarity3>(X(5),
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|
Similarity3(Rot3::Ypr(4.0 * M_PI / 2.0, 0, 0), Point3(0, 0, 0), 1.0));
|
|
|
|
//initial.print("Initial Estimate\n");
|
|
|
|
Values result;
|
|
result = LevenbergMarquardtOptimizer(graph, initial).optimize();
|
|
//result.print("Optimized Estimate\n");
|
|
Pose3 p1, p2, p3, p4, p5;
|
|
p1 = Pose3(result.at<Similarity3>(X(1)));
|
|
p2 = Pose3(result.at<Similarity3>(X(2)));
|
|
p3 = Pose3(result.at<Similarity3>(X(3)));
|
|
p4 = Pose3(result.at<Similarity3>(X(4)));
|
|
p5 = Pose3(result.at<Similarity3>(X(5)));
|
|
|
|
//p1.print("Pose1");
|
|
//p2.print("Pose2");
|
|
//p3.print("Pose3");
|
|
//p4.print("Pose4");
|
|
//p5.print("Pose5");
|
|
|
|
Similarity3 expected(0.7);
|
|
EXPECT(assert_equal(expected, result.at<Similarity3>(X(5)), 0.4));
|
|
}
|
|
|
|
//******************************************************************************
|
|
// Align points (p,q) assuming that p = T*q + noise
|
|
TEST(Similarity3, AlignScaledPointClouds) {
|
|
// Create ground truth
|
|
Point3 q1(0, 0, 0), q2(1, 0, 0), q3(0, 1, 0);
|
|
|
|
// Create transformed cloud (noiseless)
|
|
// Point3 p1 = T4 * q1, p2 = T4 * q2, p3 = T4 * q3;
|
|
|
|
// Create an unknown expression
|
|
Expression<Similarity3> unknownT(0); // use key 0
|
|
|
|
// Create constant expressions for the ground truth points
|
|
Expression<Point3> q1_(q1), q2_(q2), q3_(q3);
|
|
|
|
// Create prediction expressions
|
|
Expression<Point3> predict1(unknownT, &Similarity3::transformFrom, q1_);
|
|
Expression<Point3> predict2(unknownT, &Similarity3::transformFrom, q2_);
|
|
Expression<Point3> predict3(unknownT, &Similarity3::transformFrom, q3_);
|
|
|
|
//// Create Expression factor graph
|
|
// ExpressionFactorGraph graph;
|
|
// graph.addExpressionFactor(predict1, p1, R); // |T*q1 - p1|
|
|
// graph.addExpressionFactor(predict2, p2, R); // |T*q2 - p2|
|
|
// graph.addExpressionFactor(predict3, p3, R); // |T*q3 - p3|
|
|
}
|
|
|
|
//******************************************************************************
|
|
TEST(Similarity3 , Invariants) {
|
|
Similarity3 id;
|
|
|
|
EXPECT(check_group_invariants(id, id));
|
|
EXPECT(check_group_invariants(id, T3));
|
|
EXPECT(check_group_invariants(T2, id));
|
|
EXPECT(check_group_invariants(T2, T3));
|
|
|
|
EXPECT(check_manifold_invariants(id, id));
|
|
EXPECT(check_manifold_invariants(id, T3));
|
|
EXPECT(check_manifold_invariants(T2, id));
|
|
EXPECT(check_manifold_invariants(T2, T3));
|
|
}
|
|
|
|
//******************************************************************************
|
|
TEST(Similarity3 , LieGroupDerivatives) {
|
|
Similarity3 id;
|
|
|
|
CHECK_LIE_GROUP_DERIVATIVES(id, id);
|
|
CHECK_LIE_GROUP_DERIVATIVES(id, T2);
|
|
CHECK_LIE_GROUP_DERIVATIVES(T2, id);
|
|
CHECK_LIE_GROUP_DERIVATIVES(T2, T3);
|
|
}
|
|
|
|
//******************************************************************************
|
|
int main() {
|
|
TestResult tr;
|
|
return TestRegistry::runAllTests(tr);
|
|
}
|
|
//******************************************************************************
|
|
|