二维傅里叶变换
二维傅里叶变换
第一种做法是拆成一维的做。
#include 第二种根据二维傅里叶变换公式
[F(u,v)=∑x=0N−1∑y=0N−1f(x,y)e−j2πN(ux+vy)] [ F ( u , v ) = ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) e − j 2 π N ( u x + v y ) ]
拆成2次一维傅里叶变换。
#include
#include
#include
using namespace std;
const int N = 1000;
const double PI = acos(-1);
const double eps = 1e-10;
struct complex
{
double r,i;
complex(double _r = 0.0,double _i = 0.0)
{
r = _r; i = _i;
}
complex operator +(const complex &b)
{
return complex(r+b.r,i+b.i);
}
complex operator -(const complex &b)
{
return complex(r-b.r,i-b.i);
}
complex operator *(const complex &b)
{
return complex(r*b.r-i*b.i,r*b.i+i*b.r);
}
complex operator /(const int b)
{
return complex(r / b, i / b);
}
void Out(){
cout << "(" << (r < eps ? 0 : r) << " " << (i < eps ? 0 : i) << ")" << " ";
}
};
void change(complex y[],int len)
{
int i,j,k;
for(i = 1, j = len/2;i < len-1; i++)
{
if(i < j)swap(y[i],y[j]);
k = len/2;
while( j >= k)
{
j -= k;
k /= 2;
}
if(j < k) j += k;
}
}
void FFT(complex y[],int len,int on)
{
change(y,len);
for(int h = 2; h <= len; h <<= 1)
{
complex wn(cos(-on*2*PI/h),sin(-on*2*PI/h));
for(int j = 0;j < len;j+=h)
{
complex w(1,0);
for(int k = j;k < j+h/2;k++)
{
complex u = y[k];
complex t = w*y[k+h/2];
y[k] = u+t;
y[k+h/2] = u-t;
w = w*wn;
}
}
}
if(on == -1)
for(int i = 0;i < len;i++)
y[i] = y[i] / len;
}
void test_FFT(){
complex A[N * N], B[N * N];
int a[N], b[N];
int n;
cin >> n;
for(int i = 0; i < n; i++){
cin >> a[i];
A[i] = complex(a[i], 0);
}
FFT(A, n, 1);
for(int i = 0; i < n; i++){
A[i].Out();
}
cout << endl;
for(int i = 0; i < n; i++) b[i] = a[i];
for(int i = 0; i < n; i++){
for(int j = 0; j < n; j++){
complex X(b[j], 0), Y(cos(-2 * PI * i * j / n), sin(-2 * PI * i * j / n));
B[i] = B[i] + X * Y;
}
B[i].Out();
}
}
int a[N][N], b[N][N], c[N][N], FFT_c[N][N];
complex A[N][N], B[N][N], t[N][N];
int main()
{
int n, m, fn, fm, T;
freopen("../in.txt", "r", stdin);
cin >> T;
//T = 2;
while(T--) {
cin >> n >> m;
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
cin >> a[i][j];
}
}
cin >> fn >> fm;
for (int i = 0; i < fn; i++) {
for (int j = 0; j < fm; j++) {
cin >> b[i][j];
}
}
int tmp = max(n, m), len = 1;
while (len < 2 * tmp) len <<= 1;
for(int i = 0; i < len; i++){
for(int j = 0; j < len; j++){
A[i][j] = B[i][j] = complex(0, 0);
}
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
A[i][j] = complex(a[i][j], 0);
}
}
for (int i = 0; i < fn; i++) {
for (int j = 0; j < fm; j++) {
B[i][j] = complex(b[fn - 1 - i][fm - 1 - j], 0);
}
}
// for(int i = 0; i < len; i++){
// for(int j = 0; j < len; j++){
// A[i][j].Out();
// }
// cout <
// }
// for(int i = 0; i < len; i++){
// for(int j = 0; j < len; j++){
// B[i][j].Out();
// }
// cout <
// }
for (int i = 0; i < len; i++) {
FFT(A[i], len, 1);
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
t[i][j] = A[i][j];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
A[i][j] = t[j][i];
}
}
for (int i = 0; i < len; i++) {
FFT(A[i], len, 1);
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
t[i][j] = A[i][j];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
A[i][j] = t[j][i];
}
}
/////////////////////////////////
for (int i = 0; i < len; i++) {
FFT(B[i], len, 1);
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
t[i][j] = B[i][j];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
B[i][j] = t[j][i];
}
}
for (int i = 0; i < len; i++) {
FFT(B[i], len, 1);
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
t[i][j] = B[i][j];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
B[i][j] = t[j][i];
}
}
////////////////
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
A[i][j] = A[i][j] * B[i][j];
}
}
////////////////
for (int i = 0; i < len; i++) {
FFT(A[i], len, -1);
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
t[i][j] = A[i][j];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
A[i][j] = t[j][i];
}
}
for (int i = 0; i < len; i++) {
FFT(A[i], len, -1);
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
t[i][j] = A[i][j];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
A[i][j] = t[j][i];
}
}
for (int i = 0; i < len; i++) {
for (int j = 0; j < len; j++) {
//A[i][j].Out();
FFT_c[i][j] = A[i][j].r + 0.5;
}
//cout << endl;
}
/////////////////
for (int i = 0; i < n + fn - 1; i++) {
for (int j = 0; j < m + fm - 1; j++) {
int sum = 0;
for (int x = 0; x < fn; x++) {
for (int y = 0; y < fm; y++) {
if (i - x < 0 || j - y < 0 || i - x >= n || j - y >= m ) continue;
sum += a[i - x][j - y] * b[fn - 1 - x][fm - 1 - y];
}
}
c[i][j] = sum;
//cout << c[i][j] << " ";
}
//cout << endl;
}
int flag = 0;
for (int i = 0; i < n + fn - 1; i++) {
for (int j = 0; j < m + fm - 1; j++) {
if (c[i][j] != FFT_c[i][j]) {
flag = 1;
}
}
}
if (flag) cout << "No" << endl;
else cout << "Yes" << endl;
}
return 0;
}