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skia2/tests/WangsFormulaTest.cpp
Chris Dalton 50f5e68b7c Use the word "precision" to describe "1/tolerance" for linearization 
Bug: skia:10419
Change-Id: Ie01122ab8c9130367988fd2e264fcf4761862e89
Reviewed-on: https://skia-review.googlesource.com/c/skia/+/397656
Reviewed-by: Robert Phillips <robertphillips@google.com>
Commit-Queue: Chris Dalton <csmartdalton@google.com>
2021-04-19 17:38:35 +00:00

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/*
* Copyright 2020 Google Inc.
*
* Use of this source code is governed by a BSD-style license that can be
* found in the LICENSE file.
*/
#include "include/utils/SkRandom.h"
#include "src/core/SkGeometry.h"
#include "src/gpu/tessellate/GrWangsFormula.h"
#include "tests/Test.h"
constexpr static int kPrecision = 4; // 1/4 pixel max error.
const SkPoint kSerp[4] = {
{285.625f, 499.687f}, {411.625f, 808.188f}, {1064.62f, 135.688f}, {1042.63f, 585.187f}};
const SkPoint kLoop[4] = {
{635.625f, 614.687f}, {171.625f, 236.188f}, {1064.62f, 135.688f}, {516.625f, 570.187f}};
const SkPoint kQuad[4] = {
{460.625f, 557.187f}, {707.121f, 209.688f}, {779.628f, 577.687f}};
static float wangs_formula_quadratic_reference_impl(float precision, const SkPoint p[3]) {
float k = (2 * 1) / 8.f * precision;
return sqrtf(k * (p[0] - p[1]*2 + p[2]).length());
}
static float wangs_formula_cubic_reference_impl(float precision, const SkPoint p[4]) {
float k = (3 * 2) / 8.f * precision;
return sqrtf(k * std::max((p[0] - p[1]*2 + p[2]).length(),
(p[1] - p[2]*2 + p[3]).length()));
}
// Returns number of segments for linearized quadratic rational. This is an analogue
// to Wang's formula, taken from:
//
// J. Zheng, T. Sederberg. "Estimating Tessellation Parameter Intervals for
// Rational Curves and Surfaces." ACM Transactions on Graphics 19(1). 2000.
// See Thm 3, Corollary 1.
//
// Input points should be in projected space.
static float wangs_formula_conic_reference_impl(float precision,
const SkPoint P[3],
const float w) {
// Compute center of bounding box in projected space
float min_x = P[0].fX, max_x = min_x,
min_y = P[0].fY, max_y = min_y;
for (int i = 1; i < 3; i++) {
min_x = std::min(min_x, P[i].fX);
max_x = std::max(max_x, P[i].fX);
min_y = std::min(min_y, P[i].fY);
max_y = std::max(max_y, P[i].fY);
}
const SkPoint C = SkPoint::Make(0.5f * (min_x + max_x), 0.5f * (min_y + max_y));
// Translate control points and compute max length
SkPoint tP[3] = {P[0] - C, P[1] - C, P[2] - C};
float max_len = 0;
for (int i = 0; i < 3; i++) {
max_len = std::max(max_len, tP[i].length());
}
SkASSERT(max_len > 0);
// Compute delta = parametric step size of linearization
const float eps = 1 / precision;
const float r_minus_eps = std::max(0.f, max_len - eps);
const float min_w = std::min(w, 1.f);
const float numer = 4 * min_w * eps;
const float denom =
(tP[2] - tP[1] * 2 * w + tP[0]).length() + r_minus_eps * std::abs(1 - 2 * w + 1);
const float delta = sqrtf(numer / denom);
// Return corresponding num segments in the interval [tmin,tmax]
constexpr float tmin = 0, tmax = 1;
SkASSERT(delta > 0);
return (tmax - tmin) / delta;
}
static void for_random_matrices(SkRandom* rand, std::function<void(const SkMatrix&)> f) {
SkMatrix m;
m.setIdentity();
f(m);
for (int i = -10; i <= 30; ++i) {
for (int j = -10; j <= 30; ++j) {
m.setScaleX(std::ldexp(1 + rand->nextF(), i));
m.setSkewX(0);
m.setSkewY(0);
m.setScaleY(std::ldexp(1 + rand->nextF(), j));
f(m);
m.setScaleX(std::ldexp(1 + rand->nextF(), i));
m.setSkewX(std::ldexp(1 + rand->nextF(), (j + i) / 2));
m.setSkewY(std::ldexp(1 + rand->nextF(), (j + i) / 2));
m.setScaleY(std::ldexp(1 + rand->nextF(), j));
f(m);
}
}
}
static void for_random_beziers(int numPoints, SkRandom* rand,
std::function<void(const SkPoint[])> f,
int maxExponent = 30) {
SkASSERT(numPoints <= 4);
SkPoint pts[4];
for (int i = -10; i <= maxExponent; ++i) {
for (int j = 0; j < numPoints; ++j) {
pts[j].set(std::ldexp(1 + rand->nextF(), i), std::ldexp(1 + rand->nextF(), i));
}
f(pts);
}
}
// Ensure the optimized "*_log2" versions return the same value as ceil(std::log2(f)).
DEF_TEST(WangsFormula_log2, r) {
// Constructs a cubic such that the 'length' term in wang's formula == term.
//
// f = sqrt(k * length(max(abs(p0 - p1*2 + p2),
// abs(p1 - p2*2 + p3))));
auto setupCubicLengthTerm = [](int seed, SkPoint pts[], float term) {
memset(pts, 0, sizeof(SkPoint) * 4);
SkPoint term2d = (seed & 1) ?
SkPoint::Make(term, 0) : SkPoint::Make(.5f, std::sqrt(3)/2) * term;
seed >>= 1;
if (seed & 1) {
term2d.fX = -term2d.fX;
}
seed >>= 1;
if (seed & 1) {
std::swap(term2d.fX, term2d.fY);
}
seed >>= 1;
switch (seed % 4) {
case 0:
pts[0] = term2d;
pts[3] = term2d * .75f;
return;
case 1:
pts[1] = term2d * -.5f;
return;
case 2:
pts[1] = term2d * -.5f;
return;
case 3:
pts[3] = term2d;
pts[0] = term2d * .75f;
return;
}
};
// Constructs a quadratic such that the 'length' term in wang's formula == term.
//
// f = sqrt(k * length(p0 - p1*2 + p2));
auto setupQuadraticLengthTerm = [](int seed, SkPoint pts[], float term) {
memset(pts, 0, sizeof(SkPoint) * 3);
SkPoint term2d = (seed & 1) ?
SkPoint::Make(term, 0) : SkPoint::Make(.5f, std::sqrt(3)/2) * term;
seed >>= 1;
if (seed & 1) {
term2d.fX = -term2d.fX;
}
seed >>= 1;
if (seed & 1) {
std::swap(term2d.fX, term2d.fY);
}
seed >>= 1;
switch (seed % 3) {
case 0:
pts[0] = term2d;
return;
case 1:
pts[1] = term2d * -.5f;
return;
case 2:
pts[2] = term2d;
return;
}
};
// GrWangsFormula::cubic and ::quadratic both use rsqrt instead of sqrt for speed. Linearization
// is all approximate anyway, so as long as we are within ~1/2 tessellation segment of the
// reference value we are good enough.
constexpr static float kTessellationTolerance = 1/128.f;
for (int level = 0; level < 30; ++level) {
float epsilon = std::ldexp(SK_ScalarNearlyZero, level * 2);
SkPoint pts[4];
{
// Test cubic boundaries.
// f = sqrt(k * length(max(abs(p0 - p1*2 + p2),
// abs(p1 - p2*2 + p3))));
constexpr static float k = (3 * 2) / (8 * (1.f/kPrecision));
float x = std::ldexp(1, level * 2) / k;
setupCubicLengthTerm(level << 1, pts, x - epsilon);
float referenceValue = wangs_formula_cubic_reference_impl(kPrecision, pts);
REPORTER_ASSERT(r, std::ceil(std::log2(referenceValue)) == level);
float c = GrWangsFormula::cubic(kPrecision, pts);
REPORTER_ASSERT(r, SkScalarNearlyEqual(c/referenceValue, 1, kTessellationTolerance));
REPORTER_ASSERT(r, GrWangsFormula::cubic_log2(kPrecision, pts) == level);
setupCubicLengthTerm(level << 1, pts, x + epsilon);
referenceValue = wangs_formula_cubic_reference_impl(kPrecision, pts);
REPORTER_ASSERT(r, std::ceil(std::log2(referenceValue)) == level + 1);
c = GrWangsFormula::cubic(kPrecision, pts);
REPORTER_ASSERT(r, SkScalarNearlyEqual(c/referenceValue, 1, kTessellationTolerance));
REPORTER_ASSERT(r, GrWangsFormula::cubic_log2(kPrecision, pts) == level + 1);
}
{
// Test quadratic boundaries.
// f = std::sqrt(k * Length(p0 - p1*2 + p2));
constexpr static float k = 2 / (8 * (1.f/kPrecision));
float x = std::ldexp(1, level * 2) / k;
setupQuadraticLengthTerm(level << 1, pts, x - epsilon);
float referenceValue = wangs_formula_quadratic_reference_impl(kPrecision, pts);
REPORTER_ASSERT(r, std::ceil(std::log2(referenceValue)) == level);
float q = GrWangsFormula::quadratic(kPrecision, pts);
REPORTER_ASSERT(r, SkScalarNearlyEqual(q/referenceValue, 1, kTessellationTolerance));
REPORTER_ASSERT(r, GrWangsFormula::quadratic_log2(kPrecision, pts) == level);
setupQuadraticLengthTerm(level << 1, pts, x + epsilon);
referenceValue = wangs_formula_quadratic_reference_impl(kPrecision, pts);
REPORTER_ASSERT(r, std::ceil(std::log2(referenceValue)) == level+1);
q = GrWangsFormula::quadratic(kPrecision, pts);
REPORTER_ASSERT(r, SkScalarNearlyEqual(q/referenceValue, 1, kTessellationTolerance));
REPORTER_ASSERT(r, GrWangsFormula::quadratic_log2(kPrecision, pts) == level + 1);
}
}
auto check_cubic_log2 = [&](const SkPoint* pts) {
float f = std::max(1.f, wangs_formula_cubic_reference_impl(kPrecision, pts));
int f_log2 = GrWangsFormula::cubic_log2(kPrecision, pts);
REPORTER_ASSERT(r, SkScalarCeilToInt(std::log2(f)) == f_log2);
float c = std::max(1.f, GrWangsFormula::cubic(kPrecision, pts));
REPORTER_ASSERT(r, SkScalarNearlyEqual(c/f, 1, kTessellationTolerance));
};
auto check_quadratic_log2 = [&](const SkPoint* pts) {
float f = std::max(1.f, wangs_formula_quadratic_reference_impl(kPrecision, pts));
int f_log2 = GrWangsFormula::quadratic_log2(kPrecision, pts);
REPORTER_ASSERT(r, SkScalarCeilToInt(std::log2(f)) == f_log2);
float q = std::max(1.f, GrWangsFormula::quadratic(kPrecision, pts));
REPORTER_ASSERT(r, SkScalarNearlyEqual(q/f, 1, kTessellationTolerance));
};
SkRandom rand;
for_random_matrices(&rand, [&](const SkMatrix& m) {
SkPoint pts[4];
m.mapPoints(pts, kSerp, 4);
check_cubic_log2(pts);
m.mapPoints(pts, kLoop, 4);
check_cubic_log2(pts);
m.mapPoints(pts, kQuad, 3);
check_quadratic_log2(pts);
});
for_random_beziers(4, &rand, [&](const SkPoint pts[]) {
check_cubic_log2(pts);
});
for_random_beziers(3, &rand, [&](const SkPoint pts[]) {
check_quadratic_log2(pts);
});
}
// Ensure using transformations gives the same result as pre-transforming all points.
DEF_TEST(WangsFormula_vectorXforms, r) {
auto check_cubic_log2_with_transform = [&](const SkPoint* pts, const SkMatrix& m){
SkPoint ptsXformed[4];
m.mapPoints(ptsXformed, pts, 4);
int expected = GrWangsFormula::cubic_log2(kPrecision, ptsXformed);
int actual = GrWangsFormula::cubic_log2(kPrecision, pts, GrVectorXform(m));
REPORTER_ASSERT(r, actual == expected);
};
auto check_quadratic_log2_with_transform = [&](const SkPoint* pts, const SkMatrix& m) {
SkPoint ptsXformed[3];
m.mapPoints(ptsXformed, pts, 3);
int expected = GrWangsFormula::quadratic_log2(kPrecision, ptsXformed);
int actual = GrWangsFormula::quadratic_log2(kPrecision, pts, GrVectorXform(m));
REPORTER_ASSERT(r, actual == expected);
};
SkRandom rand;
for_random_matrices(&rand, [&](const SkMatrix& m) {
check_cubic_log2_with_transform(kSerp, m);
check_cubic_log2_with_transform(kLoop, m);
check_quadratic_log2_with_transform(kQuad, m);
for_random_beziers(4, &rand, [&](const SkPoint pts[]) {
check_cubic_log2_with_transform(pts, m);
});
for_random_beziers(3, &rand, [&](const SkPoint pts[]) {
check_quadratic_log2_with_transform(pts, m);
});
});
}
DEF_TEST(WangsFormula_worst_case_cubic, r) {
{
SkPoint worstP[] = {{0,0}, {100,100}, {0,0}, {0,0}};
REPORTER_ASSERT(r, GrWangsFormula::worst_case_cubic(kPrecision, 100, 100) ==
wangs_formula_cubic_reference_impl(kPrecision, worstP));
REPORTER_ASSERT(r, GrWangsFormula::worst_case_cubic_log2(kPrecision, 100, 100) ==
GrWangsFormula::cubic_log2(kPrecision, worstP));
}
{
SkPoint worstP[] = {{100,100}, {100,100}, {200,200}, {100,100}};
REPORTER_ASSERT(r, GrWangsFormula::worst_case_cubic(kPrecision, 100, 100) ==
wangs_formula_cubic_reference_impl(kPrecision, worstP));
REPORTER_ASSERT(r, GrWangsFormula::worst_case_cubic_log2(kPrecision, 100, 100) ==
GrWangsFormula::cubic_log2(kPrecision, worstP));
}
auto check_worst_case_cubic = [&](const SkPoint* pts) {
SkRect bbox;
bbox.setBoundsNoCheck(pts, 4);
float worst = GrWangsFormula::worst_case_cubic(kPrecision, bbox.width(), bbox.height());
int worst_log2 = GrWangsFormula::worst_case_cubic_log2(kPrecision, bbox.width(),
bbox.height());
float actual = wangs_formula_cubic_reference_impl(kPrecision, pts);
REPORTER_ASSERT(r, worst >= actual);
REPORTER_ASSERT(r, std::ceil(std::log2(std::max(1.f, worst))) == worst_log2);
};
SkRandom rand;
for (int i = 0; i < 100; ++i) {
for_random_beziers(4, &rand, [&](const SkPoint pts[]) {
check_worst_case_cubic(pts);
});
}
}
// Ensure Wang's formula for quads produces max error within tolerance.
DEF_TEST(WangsFormula_quad_within_tol, r) {
// Wang's formula and the quad math starts to lose precision with very large
// coordinate values, so limit the magnitude a bit to prevent test failures
// due to loss of precision.
constexpr int maxExponent = 15;
SkRandom rand;
for_random_beziers(3, &rand, [&r](const SkPoint pts[]) {
const int nsegs = static_cast<int>(
std::ceil(wangs_formula_quadratic_reference_impl(kPrecision, pts)));
const float tdelta = 1.f / nsegs;
for (int j = 0; j < nsegs; ++j) {
const float tmin = j * tdelta, tmax = (j + 1) * tdelta;
// Get section of quad in [tmin,tmax]
const SkPoint* sectionPts;
SkPoint tmp0[5];
SkPoint tmp1[5];
if (tmin == 0) {
if (tmax == 1) {
sectionPts = pts;
} else {
SkChopQuadAt(pts, tmp0, tmax);
sectionPts = tmp0;
}
} else {
SkChopQuadAt(pts, tmp0, tmin);
if (tmax == 1) {
sectionPts = tmp0 + 2;
} else {
SkChopQuadAt(tmp0 + 2, tmp1, (tmax - tmin) / (1 - tmin));
sectionPts = tmp1;
}
}
// For quads, max distance from baseline is always at t=0.5.
SkPoint p;
p = SkEvalQuadAt(sectionPts, 0.5f);
// Get distance of p to baseline
const SkPoint n = {sectionPts[2].fY - sectionPts[0].fY,
sectionPts[0].fX - sectionPts[2].fX};
const float d = std::abs((p - sectionPts[0]).dot(n)) / n.length();
// Check distance is within specified tolerance
REPORTER_ASSERT(r, d <= (1.f / kPrecision) + SK_ScalarNearlyZero);
}
}, maxExponent);
}
// Ensure the specialized version for rational quads reduces to regular Wang's
// formula when all weights are equal to one
DEF_TEST(WangsFormula_rational_quad_reduces, r) {
constexpr static float kTessellationTolerance = 1 / 128.f;
SkRandom rand;
for (int i = 0; i < 100; ++i) {
for_random_beziers(3, &rand, [&r](const SkPoint pts[]) {
const float rational_nsegs = wangs_formula_conic_reference_impl(kPrecision, pts, 1.f);
const float integral_nsegs = wangs_formula_quadratic_reference_impl(kPrecision, pts);
REPORTER_ASSERT(
r, SkScalarNearlyEqual(rational_nsegs, integral_nsegs, kTessellationTolerance));
});
}
}
// Ensure the rational quad version (used for conics) produces max error within tolerance.
DEF_TEST(WangsFormula_conic_within_tol, r) {
constexpr int maxExponent = 24;
// Single-precision functions in SkConic/SkGeometry lose too much accuracy with
// large-magnitude curves and large weights for this test to pass.
using Sk2d = skvx::Vec<2, double>;
const auto eval_conic = [](const SkPoint pts[3], float w, float t) -> Sk2d {
const auto eval = [](Sk2d A, Sk2d B, Sk2d C, float t) -> Sk2d {
return (A * t + B) * t + C;
};
const Sk2d p0 = {pts[0].fX, pts[0].fY};
const Sk2d p1 = {pts[1].fX, pts[1].fY};
const Sk2d p1w = p1 * w;
const Sk2d p2 = {pts[2].fX, pts[2].fY};
Sk2d numer = eval(p2 - p1w * 2 + p0, (p1w - p0) * 2, p0, t);
Sk2d denomC = {1, 1};
Sk2d denomB = {2 * (w - 1), 2 * (w - 1)};
Sk2d denomA = {-2 * (w - 1), -2 * (w - 1)};
Sk2d denom = eval(denomA, denomB, denomC, t);
return numer / denom;
};
const auto dot = [](const Sk2d& a, const Sk2d& b) -> double {
return a[0] * b[0] + a[1] * b[1];
};
const auto length = [](const Sk2d& p) -> double { return sqrt(p[0] * p[0] + p[1] * p[1]); };
SkRandom rand;
for (int i = -10; i <= 10; ++i) {
const float w = std::ldexp(1 + rand.nextF(), i);
for_random_beziers(
3, &rand,
[&](const SkPoint pts[]) {
const int nsegs = static_cast<int>(
std::ceil(wangs_formula_conic_reference_impl(kPrecision, pts, w)));
const float tdelta = 1.f / nsegs;
for (int j = 0; j < nsegs; ++j) {
const float tmin = j * tdelta, tmax = (j + 1) * tdelta,
tmid = 0.5f * (tmin + tmax);
Sk2d p0, p1, p2;
p0 = eval_conic(pts, w, tmin);
p1 = eval_conic(pts, w, tmid);
p2 = eval_conic(pts, w, tmax);
// Get distance of p1 to baseline (p0, p2).
const Sk2d n = {p2[1] - p0[1], p0[0] - p2[0]};
SkASSERT(length(n) != 0);
const double d = std::abs(dot(p1 - p0, n)) / length(n);
// Check distance is within tolerance
REPORTER_ASSERT(r, d <= (1.0 / kPrecision) + SK_ScalarNearlyZero);
}
},
maxExponent);
}
}
// Ensure the vectorized conic version equals the reference implementation
DEF_TEST(WangsFormula_conic_matches_reference, r) {
constexpr static float kTolerance = 1.f / kPrecision;
SkRandom rand;
for (int i = -10; i <= 10; ++i) {
const float w = std::ldexp(1 + rand.nextF(), i);
for_random_beziers(3, &rand, [&r, w](const SkPoint pts[]) {
const float ref_nsegs = wangs_formula_conic_reference_impl(kPrecision, pts, w);
const float nsegs = GrWangsFormula::conic(kTolerance, pts, w);
// Because the Gr version may implement the math differently for performance,
// allow different slack in the comparison based on the rough scale of the answer.
const float cmpThresh = ref_nsegs * (1.f / (1 << 20));
REPORTER_ASSERT(r, SkScalarNearlyEqual(ref_nsegs, nsegs, cmpThresh));
});
}
}
// Ensure using transformations gives the same result as pre-transforming all points.
DEF_TEST(WangsFormula_conic_vectorXforms, r) {
constexpr static float kTolerance = 1.f / kPrecision;
auto check_conic_with_transform = [&](const SkPoint* pts, float w, const SkMatrix& m) {
SkPoint ptsXformed[3];
m.mapPoints(ptsXformed, pts, 3);
float expected = GrWangsFormula::conic(kTolerance, ptsXformed, w);
float actual = GrWangsFormula::conic(kTolerance, pts, w, GrVectorXform(m));
REPORTER_ASSERT(r, SkScalarNearlyEqual(actual, expected));
};
SkRandom rand;
for (int i = -10; i <= 10; ++i) {
const float w = std::ldexp(1 + rand.nextF(), i);
for_random_beziers(3, &rand, [&](const SkPoint pts[]) {
check_conic_with_transform(pts, w, SkMatrix::I());
check_conic_with_transform(
pts, w, SkMatrix::Scale(rand.nextRangeF(-10, 10), rand.nextRangeF(-10, 10)));
// Random 2x2 matrix
SkMatrix m;
m.setScaleX(rand.nextRangeF(-10, 10));
m.setSkewX(rand.nextRangeF(-10, 10));
m.setSkewY(rand.nextRangeF(-10, 10));
m.setScaleY(rand.nextRangeF(-10, 10));
check_conic_with_transform(pts, w, m);
});
}
}
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