Fast Image Match

Time Limit: 10 Seconds Memory Limit: 32768 KB

Given two images A and B, use image B to cover image A. Where would we put B on A, so that the overlapping part of A and B has the most likelihood? To simplify the problem, we assume that A and B only contain numbers between 0 and 255. The difference between A and B is defined as the square sum of the differences of corresponding elements in the overlapped parts of A and B.

For example, we have

A (3 * 3):	a1	a2	a3	B (2 * 2):	b1	b2
	a4	a5	a6		b4	b5
	a7	a8	a9

When B is placed on position a5, the difference of them is ((b1-a5)^2 + (b2-a6)^2 + (b4-a8)^2 + (b5-a9)^2). Now we hope to have the position of the top left corner of B that gives the minimum difference. (B must completely reside on A)

It is clear that a simple solution will appear with very low efficiency when A and B have too many elements. But we can use 1-dimensional repeat convolution, which can be computed by Fast Fourier Transform (FFT), to improve the performance.

A program with explanation of FFT is given below:


/**
 * Given two sequences {a1, a2, a3.. an} and {b1, b2, b3... bn},
 * their repeat convolution means:
 * r1 = a1*b1 + a2*b2 + a3*b3 + ... + an*bn
 * r2 = a1*bn + a2*b1 + a3*b2 + ... + an*bn-1
 * r3 = a1*bn-1 + a2*bn + a3*b1 + ... + an*bn-2
 * ...
 * rn = a1*b2 + a2*b3 + a3*b4 + ... + an-1*bn + an*b1
 * Notice n >= 2 and n must be power of 2.
 */
#include <vector>
#include <complex>
#include <cmath>
#define for if (0); else for
using namespace std;

const int MaxFastBits = 16;
int **gFFTBitTable = 0;

int NumberOfBitsNeeded(int PowerOfTwo) {
 for (int i = 0;; ++i) {
 if (PowerOfTwo & (1 << i)) {
 return i;
 }
 }
}

int ReverseBits(int index, int NumBits) {
 int ret = 0;
 for (int i = 0; i < NumBits; ++i, index >>= 1) {
 ret = (ret << 1) | (index & 1);
 }
 return ret;
}

void InitFFT() {
 gFFTBitTable = new int *[MaxFastBits];
 for (int i = 1, length = 2; i <= MaxFastBits; ++i, length <<= 1) {
 gFFTBitTable[i - 1] = new int[length];
 for (int j = 0; j < length; ++j) {
 gFFTBitTable[i - 1][j] = ReverseBits(j, i);
 }
 }
}
inline int FastReverseBits(int i, int NumBits) {
 return NumBits <= MaxFastBits ? gFFTBitTable[NumBits - 1][i] : ReverseBits(i, NumBits);
}

void FFT(bool InverseTransform, vector<complex<double> >& In, vector<complex<double> >& Out) {
 if (!gFFTBitTable) { InitFFT(); }
 // simultaneous data copy and bit-reversal ordering into outputs
 int NumSamples = In.size();
 int NumBits = NumberOfBitsNeeded(NumSamples);
 for (int i = 0; i < NumSamples; ++i) {
 Out[FastReverseBits(i, NumBits)] = In[i];
 }
 // the FFT process
 double angle_numerator = acos(-1.) * (InverseTransform ? -2 : 2);
 for (int BlockEnd = 1, BlockSize = 2; BlockSize <= NumSamples; BlockSize <<= 1) {
 double delta_angle = angle_numerator / BlockSize;
 double sin1 = sin(-delta_angle);
 double cos1 = cos(-delta_angle);
 double sin2 = sin(-delta_angle * 2);
 double cos2 = cos(-delta_angle * 2);
 for (int i = 0; i < NumSamples; i += BlockSize) {
 complex<double> a1(cos1, sin1), a2(cos2, sin2);
 for (int j = i, n = 0; n < BlockEnd; ++j, ++n) {
 complex<double> a0(2 * cos1 * a1.real() - a2.real(), 2 * cos1 * a1.imag() - a2.imag());
 a2 = a1;
 a1 = a0;
 complex<double> a = a0 * Out[j + BlockEnd];
 Out[j + BlockEnd] = Out[j] - a;
 Out[j] += a;
 }
 }
 BlockEnd = BlockSize;
 }
 // normalize if inverse transform
 if (InverseTransform) {
 for (int i = 0; i < NumSamples; ++i) {
 Out[i] /= NumSamples;
 }
 }
}

vector<double> convolution(vector<double> a, vector<double> b) {
 int n = a.size();
 vector<complex<double> > s(n), d1(n), d2(n), y(n);
 for (int i = 0; i < n; ++i) {
 s[i] = complex<double>(a[i], 0);
 }
 FFT(false, s, d1);
 s[0] = complex<double>(b[0], 0);
 for (int i = 1; i < n; ++i) {
 s[i] = complex<double>(b[n - i], 0);
 }
 FFT(false, s, d2);
 for (int i = 0; i < n; ++i) {
 y[i] = d1[i] * d2[i];
 }
 FFT(true, y, s);
 vector<double> ret(n);
 for (int i = 0; i < n; ++i) {
 ret[i] = s[i].real();
 }
 return ret;
}

int main() {
 double a[4] = {1, 2, 3, 4}, b[4] = {1, 2, 3, 4};
 vector<double> r = convolution(vector<double>(a, a + 4), vector<double>(b, b + 4));
 // r[0] = 30 (1*1 + 2*2 + 3*3 + 4*4)
 // r[1] = 24 (1*4 + 2*1 + 3*2 + 4*3)
 // r[2] = 22 (1*3 + 2*4 + 3*1 + 4*2)
 // r[3] = 24 (1*2 + 2*3 + 3*4 + 4*1)
 return 0;
}

Input

The first line contains n (1 <= n <= 10), the number of test cases.

For each test case, the first line contains four integers m, n, p and q, where A is a matrix of m * n, B is a matrix of p * q (2 <= m, n, p, q <= 500, m >= p, n >= q). The following m lines are the elements of A and p lines are the elements of B.

Output

For each case, print the position that gives the minimum difference (the top left corner of A is (1, 1)). You can assume that each test case has a unique solution.

Sample Input

Sample Output

1 1
1 2

Submit

Source: Zhejiang University Local Contest 2003