首先，我必须承认我同意generic_opto_guy https://stackoverflow.com/users/9400869/generic-opto-guy:
循环方法看起来不错，所以我们需要检查数学。我注意到的事情是： if (oldRow > 0 && oldCol > 0 && oldRow
尽管如此，我还是忍不住回答。 （也许，这只是我的一个形象阶段。）
我建议使用矩阵变换，而不是摆弄 sin() 和 cos()。乍一看，这可能显得过度设计，但后来您会发现它具有更大的灵活性。使用变换矩阵，您可以表达多种变换（平移、旋转、缩放、剪切、投影）以及将多个变换组合到一个矩阵中。
（可能发生的事情的预告片：SO：如何在 2D 中绘制/变形 QImage？ https://stackoverflow.com/a/56970955/7478597)
在图像中，像素可以通过二维坐标来寻址。因此，我们想到了 2×2 矩阵，但 2×2 矩阵无法表达平移。为了这，齐次坐标 https://en.wikipedia.org/wiki/Homogeneous_coordinates引入了一种数学技巧，通过将维度扩展一来处理同一空间中的位置和方向。
简而言之，二维位置 (x, y) 具有齐次坐标 (x, y, 1)。
用变换矩阵变换的位置：
v´ = M · v.
This may or may not change the value of third component. To convert the homogeneous coordinate to 2D position again, x and y has to be divided by 3rd component.
Vec2 transform(const Mat3x3 &mat, const Vec2 &pos)
{
const Vec3 pos_ = mat * Vec3(pos, 1.0);
return Vec2(pos_.x / pos_.z, pos_.y / pos_.z);
}
要将源图像转换为目标图像，可以使用以下函数：
void transform(
const Image &imgSrc, const Mat3x3 &mat, Image &imgDst,
int rgbFail = 0x808080)
{
const Mat3x3 matInv = invert(mat);
for (int y = 0; y < imgDst.h(); ++y) {
for (int x = 0; x < imgDst.w(); ++x) {
const Vec2 pos = transform(matInv, Vec2(x, y));
const int xSrc = (int)(pos.x + 0.5), ySrc = (int)(pos.y + 0.5);
imgDst.setPixel(x, y,
xSrc >= 0 && xSrc < imgSrc.w() && ySrc >= 0 && ySrc < imgSrc.h()
? imgSrc.getPixel(xSrc, ySrc)
: rgbFail);
}
}
}
Note:
变换矩阵mat描述源图像坐标到目标图像坐标的转换。嵌套循环迭代目标图像。因此，必须使用逆矩阵（表示逆变换）来获取映射到当前目标坐标的相应源图像坐标。
…以及旋转的矩阵构造函数：
enum ArgInitRot { InitRot };
template <typename VALUE>
struct Mat3x3T {
union {
VALUE comp[3 * 3];
struct {
VALUE _00, _01, _02;
VALUE _10, _11, _12;
VALUE _20, _21, _22;
};
};
// constructor to build a matrix for rotation
Mat3x3T(ArgInitRot, VALUE angle):
_00(std::cos(angle)), _01(-std::sin(angle)), _02((VALUE)0),
_10(std::sin(angle)), _11( std::cos(angle)), _12((VALUE)0),
_20( (VALUE)0), _21( (VALUE)0), _22((VALUE)1)
{ }
可以用来构造一个旋转angle（以度为单位）：
Mat3x3T<double> mat(InitRot, degToRad(30.0));
Note:
我想强调一下变换后的坐标是如何使用的：
const Vec2 pos = transform(matInv, Vec2(x, y));
const int xSrc = (int)(pos.x + 0.5), ySrc = (int)(pos.y + 0.5);
对结果进行四舍五入以产生一个离散像素位置实际上就是所谓的最近邻。或者，现在丢弃的小数部分可以用于相邻像素之间的线性插值。
为了制作一个小样本，我首先复制了image.h, image.cc, imagePPM.h, and imagePPM.cc https://stackoverflow.com/a/56850226/7478597来自我最近写的另一个答案。 （这PPM 文件格式 https://en.wikipedia.org/wiki/Netpbm_format已被使用，因为它需要最少的文件 I/O 代码。）
接下来，我用了linMath.h https://github.com/scheff173/NoGL3dDemo/blob/master/linmath.h（我的 3D 变换的最小数学集合）为 2D 变换创建一个最小的数学集合 –linMath.h:
#ifndef LIN_MATH_H
#define LIN_MATH_H
#include <iostream>
#include <cassert>
#include <cmath>
extern const double Pi;
template <typename VALUE>
inline VALUE degToRad(VALUE angle)
{
return (VALUE)Pi * angle / (VALUE)180;
}
template <typename VALUE>
inline VALUE radToDeg(VALUE angle)
{
return (VALUE)180 * angle / (VALUE)Pi;
}
enum ArgNull { Null };
template <typename VALUE>
struct Vec2T {
typedef VALUE Value;
Value x, y;
// default constructor (leaving elements uninitialized)
Vec2T() { }
Vec2T(ArgNull): x((Value)0), y((Value)0) { }
Vec2T(Value x, Value y): x(x), y(y) { }
};
typedef Vec2T<float> Vec2f;
typedef Vec2T<double> Vec2;
template <typename VALUE>
struct Vec3T {
typedef VALUE Value;
Value x, y, z;
// default constructor (leaving elements uninitialized)
Vec3T() { }
Vec3T(ArgNull): x((Value)0), y((Value)0), z((Value)0) { }
Vec3T(Value x, Value y, Value z): x(x), y(y), z(z) { }
Vec3T(const Vec2T<Value> &xy, Value z): x(xy.x), y(xy.y), z(z) { }
explicit operator Vec2T<Value>() const { return Vec2T<Value>(x, y); }
const Vec2f xy() const { return Vec2f(x, y); }
const Vec2f xz() const { return Vec2f(x, z); }
const Vec2f yz() const { return Vec2f(y, z); }
};
typedef Vec3T<float> Vec3f;
typedef Vec3T<double> Vec3;
enum ArgInitIdent { InitIdent };
enum ArgInitTrans { InitTrans };
enum ArgInitRot { InitRot };
enum ArgInitScale { InitScale };
enum ArgInitFrame { InitFrame };
template <typename VALUE>
struct Mat3x3T {
union {
VALUE comp[3 * 3];
struct {
VALUE _00, _01, _02;
VALUE _10, _11, _12;
VALUE _20, _21, _22;
};
};
// default constructor (leaving elements uninitialized)
Mat3x3T() { }
// constructor to build a matrix by elements
Mat3x3T(
VALUE _00, VALUE _01, VALUE _02,
VALUE _10, VALUE _11, VALUE _12,
VALUE _20, VALUE _21, VALUE _22):
_00(_00), _01(_01), _02(_02),
_10(_10), _11(_11), _12(_12),
_20(_20), _21(_21), _22(_22)
{ }
// constructor to build an identity matrix
Mat3x3T(ArgInitIdent):
_00((VALUE)1), _01((VALUE)0), _02((VALUE)0),
_10((VALUE)0), _11((VALUE)1), _12((VALUE)0),
_20((VALUE)0), _21((VALUE)0), _22((VALUE)1)
{ }
// constructor to build a matrix for translation
Mat3x3T(ArgInitTrans, const Vec2T<VALUE> &t):
_00((VALUE)1), _01((VALUE)0), _02((VALUE)t.x),
_10((VALUE)0), _11((VALUE)1), _12((VALUE)t.y),
_20((VALUE)0), _21((VALUE)0), _22((VALUE)1)
{ }
// constructor to build a matrix for rotation
Mat3x3T(ArgInitRot, VALUE angle):
_00(std::cos(angle)), _01(-std::sin(angle)), _02((VALUE)0),
_10(std::sin(angle)), _11( std::cos(angle)), _12((VALUE)0),
_20( (VALUE)0), _21( (VALUE)0), _22((VALUE)1)
{ }
// constructor to build a matrix for translation/rotation
Mat3x3T(ArgInitFrame, const Vec2T<VALUE> &t, VALUE angle):
_00(std::cos(angle)), _01(-std::sin(angle)), _02((VALUE)t.x),
_10(std::sin(angle)), _11( std::cos(angle)), _12((VALUE)t.y),
_20( (VALUE)0), _21( (VALUE)0), _22((VALUE)1)
{ }
// constructor to build a matrix for scaling
Mat3x3T(ArgInitScale, VALUE sx, VALUE sy):
_00((VALUE)sx), _01( (VALUE)0), _02((VALUE)0),
_10( (VALUE)0), _11((VALUE)sy), _12((VALUE)0),
_20( (VALUE)0), _21( (VALUE)0), _22((VALUE)1)
{ }
// operator to allow access with [][]
VALUE* operator [] (int i)
{
assert(i >= 0 && i < 3);
return comp + 3 * i;
}
// operator to allow access with [][]
const VALUE* operator [] (int i) const
{
assert(i >= 0 && i < 3);
return comp + 3 * i;
}
// multiply matrix with matrix -> matrix
Mat3x3T operator * (const Mat3x3T &mat) const
{
return Mat3x3T(
_00 * mat._00 + _01 * mat._10 + _02 * mat._20,
_00 * mat._01 + _01 * mat._11 + _02 * mat._21,
_00 * mat._02 + _01 * mat._12 + _02 * mat._22,
_10 * mat._00 + _11 * mat._10 + _12 * mat._20,
_10 * mat._01 + _11 * mat._11 + _12 * mat._21,
_10 * mat._02 + _11 * mat._12 + _12 * mat._22,
_20 * mat._00 + _21 * mat._10 + _22 * mat._20,
_20 * mat._01 + _21 * mat._11 + _22 * mat._21,
_20 * mat._02 + _21 * mat._12 + _22 * mat._22);
}
// multiply matrix with vector -> vector
Vec3T<VALUE> operator * (const Vec3T<VALUE> &vec) const
{
return Vec3T<VALUE>(
_00 * vec.x + _01 * vec.y + _02 * vec.z,
_10 * vec.x + _11 * vec.y + _12 * vec.z,
_20 * vec.x + _21 * vec.y + _22 * vec.z);
}
};
typedef Mat3x3T<float> Mat3x3f;
typedef Mat3x3T<double> Mat3x3;
template <typename VALUE>
std::ostream& operator<<(std::ostream &out, const Mat3x3T<VALUE> &m)
{
return out
<< m._00 << '\t' << m._01 << '\t' << m._02 << '\n'
<< m._10 << '\t' << m._11 << '\t' << m._12 << '\n'
<< m._20 << '\t' << m._21 << '\t' << m._22 << '\n';
}
/* computes determinant of a matrix.
*
* det = |M|
*
* mat ... the matrix
*/
template <typename VALUE>
VALUE determinant(const Mat3x3T<VALUE> &mat)
{
return mat._00 * mat._11 * mat._22
+ mat._01 * mat._12 * mat._20
+ mat._02 * mat._10 * mat._21
- mat._20 * mat._11 * mat._02
- mat._21 * mat._12 * mat._00
- mat._22 * mat._10 * mat._01;
}
/* returns the inverse of a regular matrix.
*
* mat matrix to invert
* eps epsilon for regularity of matrix
*/
template <typename VALUE>
Mat3x3T<VALUE> invert(
const Mat3x3T<VALUE> &mat, VALUE eps = (VALUE)1E-10)
{
assert(eps >= (VALUE)0);
// compute determinant and check that it its unequal to 0
// (Otherwise, matrix is singular!)
const VALUE det = determinant(mat);
if (std::abs(det) < eps) throw std::domain_error("Singular matrix!");
// reciproke of determinant
const VALUE detInvPos = (VALUE)1 / det, detInvNeg = -detInvPos;
// compute each element by determinant of sub-matrix which is build
// striking out row and column of pivot element itself
// BTW, the determinant is multiplied with -1 when sum of row and column
// index is odd (chess board rule)
// (This is usually called cofactor of related element.)
// transpose matrix and multiply with 1/determinant of original matrix
return Mat3x3T<VALUE>(
detInvPos * (mat._11 * mat._22 - mat._12 * mat._21),
detInvNeg * (mat._01 * mat._22 - mat._02 * mat._21),
detInvPos * (mat._01 * mat._12 - mat._02 * mat._11),
detInvNeg * (mat._10 * mat._22 - mat._12 * mat._20),
detInvPos * (mat._00 * mat._22 - mat._02 * mat._20),
detInvNeg * (mat._00 * mat._12 - mat._02 * mat._10),
detInvPos * (mat._10 * mat._21 - mat._11 * mat._20),
detInvNeg * (mat._00 * mat._21 - mat._01 * mat._20),
detInvPos * (mat._00 * mat._11 - mat._01 * mat._10));
}
#endif // LIN_MATH_H
和定义Pi in linMath.cc:
#include "linmath.h"
const double Pi = 3.1415926535897932384626433832795;
有了所有可用的工具，我制作了示例应用程序xformRGBImg.cc:
#include <iostream>
#include <fstream>
#include <sstream>
#include <string>
#include "linMath.h"
#include "image.h"
#include "imagePPM.h"
typedef unsigned int uint;
struct Error {
const std::string text;
Error(const char *text): text(text) { }
};
const char* readArg(int &i, int argc, char **argv)
{
++i;
if (i >= argc) throw Error("Missing argument!");
return argv[i];
}
uint readArgUInt(int &i, int argc, char **argv)
{
const char *arg = readArg(i, argc, argv); char *end;
const unsigned long value = strtoul(arg, &end, 0);
if (arg == end || *end) throw Error("Unsigned integer value expected!");
if ((uint)value != value) throw Error("Unsigned integer overflow!");
return (uint)value;
}
double readArgDouble(int &i, int argc, char **argv)
{
const char *arg = readArg(i, argc, argv); char *end;
const double value = strtod(arg, &end);
if (arg == end || *end) throw Error("Floating point value expected!");
return value;
}
std::pair<uint, uint> resize(int &i, int argc, char **argv)
{
const uint w = readArgUInt(i, argc, argv);
const uint h = readArgUInt(i, argc, argv);
return std::make_pair(w, h);
}
Mat3x3 translate(int &i, int argc, char **argv)
{
const double x = readArgDouble(i, argc, argv);
const double y = readArgDouble(i, argc, argv);
return Mat3x3(InitTrans, Vec2(x, y));
}
Mat3x3 rotate(int &i, int argc, char **argv)
{
const double angle = readArgDouble(i, argc, argv);
return Mat3x3(InitRot, degToRad(angle));
}
Mat3x3 scale(int &i, int argc, char **argv)
{
const double x = readArgDouble(i, argc, argv);
const double y = readArgDouble(i, argc, argv);
return Mat3x3(InitScale, x, y);
}
Vec2 transform(const Mat3x3 &mat, const Vec2 &pos)
{
const Vec3 pos_ = mat * Vec3(pos, 1.0);
return Vec2(pos_.x / pos_.z, pos_.y / pos_.z);
}
void transform(
const Image &imgSrc, const Mat3x3 &mat, Image &imgDst,
int rgbFail = 0x808080)
{
const Mat3x3 matInv = invert(mat);
for (int y = 0; y < imgDst.h(); ++y) {
for (int x = 0; x < imgDst.w(); ++x) {
const Vec2 pos = transform(matInv, Vec2(x, y));
const int xSrc = (int)(pos.x + 0.5), ySrc = (int)(pos.y + 0.5);
imgDst.setPixel(x, y,
xSrc >= 0 && xSrc < imgSrc.w() && ySrc >= 0 && ySrc < imgSrc.h()
? imgSrc.getPixel(xSrc, ySrc)
: rgbFail);
}
}
}
const char *const usage =
"Usage:\n"
" xformRGBImg IN_FILE OUT_FILE [[CMD]...]\n"
"\n"
"Commands:\n"
" resize W H\n"
" translate X Y\n"
" rotate ANGLE\n"
" scale SX SY\n";
int main(int argc, char **argv)
{
// read command line arguments
if (argc <= 2) {
std::cerr << "Missing arguments!\n";
std::cout << usage;
return 1;
}
const std::string inFile = argv[1];
const std::string outFile = argv[2];
std::pair<uint, uint> sizeOut(0, 0);
Mat3x3 mat(InitIdent);
for (int i = 3; i < argc; ++i) try {
const std::string cmd = argv[i];
if (cmd == "resize") sizeOut = resize(i, argc, argv);
else if (cmd == "translate") mat = translate(i, argc, argv) * mat;
else if (cmd == "rotate") mat = rotate(i, argc, argv) * mat;
else if (cmd == "scale") mat = scale(i, argc, argv) * mat;
else {
std::cerr << "Wrong command!\n";
std::cout << usage;
return 1;
}
} catch (const Error &error) {
std::cerr << "Wrong argument at $" << i << "\n"
<< error.text << '\n';
std::cout << usage;
return 1;
}
// read image
Image imgSrc;
{ std::ifstream fIn(inFile.c_str(), std::ios::binary);
if (!readPPM(fIn, imgSrc)) {
std::cerr << "Reading '" << inFile << "' failed!\n";
return 1;
}
}
// set output image size
if (sizeOut.first * sizeOut.second == 0) {
sizeOut = std::make_pair(imgSrc.w(), imgSrc.h());
}
// transform image
Image imgDst;
imgDst.resize(sizeOut.first, sizeOut.second, 3 * sizeOut.second);
transform(imgSrc, mat, imgDst);
// write image
{ std::ofstream fOut(outFile.c_str(), std::ios::binary);
if (!writePPM(fOut, imgDst) || (fOut.close(), !fOut.good())) {
std::cerr << "Writing '" << outFile << "' failed!\n";
return 1;
}
}
// done
return 0;
}
Note:
命令行参数按顺序处理。每个变换命令从单位矩阵开始从左乘到已经组合的变换矩阵。这是因为变换的串联会导致矩阵的逆序乘法。 （矩阵乘法是右结合的。）
例如。对应的矩阵转换:
x' = 翻译(x)
x" = rotate(x')
x"' = scale(x")
which is
x"' = scale(rotate(翻译(x)))
is
Mtransform = Mscale · Mrotate · Mtranslate
and
x"' = Mscale · Mrotate · Mtranslate · x = Mtransform · x
编译并测试于cygwin http://www.cygwin.org:
$ g++ -std=c++11 -o xformRGBImg image.cc imagePPM.cc linMath.cc xformRGBImg.cc
$ ./xformRGBImg
Missing arguments!
Usage:
xformRGBImg IN_FILE OUT_FILE [[CMD]...]
Commands:
resize W H
translate X Y
rotate ANGLE
scale SX SY
$
最后是示例图像cat.jpg（转换成PPM https://en.wikipedia.org/wiki/Netpbm_format in GIMP https://www.gimp.org/):
尺寸为300×300。
Note:
所有嵌入图像均从 PPM 转换为 JPEG（格式为GIMP https://www.gimp.org/再次）。 （图片上传不支持PPM，也无法想象任何浏览器都能正常显示。）
从最低限度开始：
$ ./xformRGBImg cat.ppm cat.copy.ppm
$
它看起来像原来的——身份转换应该是这样的。
现在，旋转 30°：
$ ./xformRGBImg cat.ppm cat.rot30.ppm rotate 30
$
要绕某个中心旋转，就有一个 resp。前后需要翻译：
$ ./xformRGBImg cat.ppm cat.rot30c150,150.ppm \
translate -150 -150 rotate 30 translate 150 150
$
输出图像可以使用 w · √2 × h · √2 调整大小以适应任何中心旋转。
因此，输出图像的大小调整为 425 × 425，其中最后的平移分别调整为translate 212.5 212.5:
$ ./xformRGBImg cat.ppm cat.rot30c150,150.425x425.ppm \
resize 425 425 translate -150 -150 rotate 30 translate 212.5 212.5
$
缩放比例尚未检查：
$ ./xformRGBImg cat.ppm cat.rot30c150,150s0.7,0.7.ppm \
translate -150 -150 rotate 30 scale 0.7 0.7 translate 150 150
$
最后，公平地说，我想提一下我的小玩具工具的“大哥”：图像魔术师 https://imagemagick.org/index.php.
