キーワード：フーリエ変換、離散フーリエ変換、オイラーの公式、三角関数、対称性
１次元の DFT・IDFT の考え方とデータ操作方法
２次元の画像データに対する DFT・IDFT の考え方（Step1、横１行分の処理を縦方向分繰り返す）
２次元の画像データに対する DFT・IDFT の考え方（Step2、縦１行分の処理を横方向分繰り返す）
２次元の画像データに対する DFT・IDFT 操作を行う C 言語の ソースコードを掲載。
画像データは既に、複素数型（COMPLEX）の実部に登録されている事が前提（虚部は 0で初期化済み）
enum { DFT, IDFT }; //#define M_PI 3.14159265358979323846
typedef struct COMPLEX { double r, i; } COMPLEX; // Define the new COMPLEX type. There's no entity yet.
#define RotorCOMPLEX(a) (COMPLEX){ cos(a), sin(a) } // IDFT: e j(2Πai/elements), DFT: e -j(2Πai/elements)
#define AddCOMPLEX(a,b) (COMPLEX){ (a).r + (b).r, (a).i + (b).i }
#define MulCOMPLEX(a,b) (COMPLEX){ (a).r*(b).r - (a).i*(b).i, (a).r*(b).i + (a).i*(b).r }
#define ImeDivCOMPLEX(a,b)(COMPLEX){ (a).r / (b), (a).i / (b) }
#define ImeMulCOMPLEX(a,b)(COMPLEX){ (a).r * (b), (a).i * (b) }
void dft_TwoDimensional( COMPLEX **d, int lx, int ly, int flag ) {
int x, y, u, v, elements;
double unit;
COMPLEX t[(lx>ly?lx:ly)]; // See REMARK
elements = lx; unit = (flag==IDFT) ? 2*M_PI/elements : -2*M_PI/elements;
for( y=0; y<ly; y++ ) { // Σdft_oneDimensional( for Y axis ...);
for( u=0; u<elements; u++ ) t[u]=(COMPLEX){0,0}; // From 0 To LX -1. Initialize the work area with 0.
for( u=0; u<elements; u++ ) {// "for(v=0;v<elements;v++)" is performing a convolutional operation.
for( v=0; v<elements; v++ ) t[u] = AddCOMPLEX( t[u], MulCOMPLEX(d[y][v],RotorCOMPLEX((unit*u)*v)) );
}
if(flag==IDFT) for( u=0; u<elements; u++ ) d[y][u] = ImeDivCOMPLEX(t[u],elements);
else for( u=0; u<elements; u++ ) d[y][u] = t[u];
} //////////////////////////////////////////////////////////////////////////
elements = ly; unit = (flag==IDFT) ? 2*M_PI/elements : -2*M_PI/elements;
for( x=0; x<lx; x++ ) { // Σdft_oneDimensional( for X axis ...);
for( u=0; u<elements; u++ ) t[u]=(COMPLEX){0,0}; // From 0 To LY -1. Initialize the work area with 0.
for( u=0; u<elements; u++ ) {// "for(v=0;v<elements;v++)" is performing a convolutional operation.
for( v=0; v<elements; v++ ) t[u] = AddCOMPLEX( t[u], MulCOMPLEX(d[v][x],RotorCOMPLEX((unit*u)*v)) );
}
if(flag==IDFT) for( u=0; u<elements; u++ ) d[u][x] = ImeDivCOMPLEX(t[u],elements);
else for( u=0; u<elements; u++ ) d[u][x] = t[u];
} //////////////////////////////////////////////////////////////////////////
}順変換 逆変換の間で、周波数空間でのフィルター処理を行う場合は、本点を考慮する事
#define AppendCOMPLEX(a,b) { (a).r += (b).r, (a).i += (b).i ;} も利用可
Ｃ言語の難しさかは、同じ事を複数の記述が可能な表現力の自由さに起因するのかも
// REMARK
// Compare lx and ly and use the larger number as the reserved area size.
// This memory dynamic allocation method does not require free().
// It's allocated in stack area, so you should care stack overflow.
// t[u] = (COMPLEX){0,0}; /*構造体への一括代入・初期化 古典的記述方法*/
// キャスト演算子を用いて、データセットの属性(型)を代入先と合わせることが必須条件
// Classic description method: Conform to the ISO 1990 C standard, known as c89.
// At the time of execution, data is collectively assigned, or initialize, to
// the structure. A cast operation is required for synchronizing object type.
// As an element of the structure, the integer 0 is automatically converted to
// the real 0.0. (COMPLEX){0,0}=>(COMPLEX){0.,0.}=>t[u] Even gcc 3.x works well.
// "COMPLEX **d, int lx, int ly" same as "COMPLEX d[y][x], {0<=y<ly,0<=x<lx}"
// If the two-dimensional array was generated by the following way.
// COMPLEX **d = (COMPLEX **)alloc2Darray( sizeof(COMPLEX), lx, ly );
// Finally, you need to free the allocated memory pointer, like the following "free(d);".
// The worth and positioning of the structure
// Defining a structure in C is the same as defining a database field in a database.
// Defining the simplest and most effective primitives in building a system is equi-
// valent to a complete understanding of the features and requirements of the system.
// In addition, 'smart' expressions are required in terms of notation and operation,
// which are highly readable and less prone to errors.
// After learning the basic grammar of C language, it is need to think about the
// gathering elements, the operation method may be included, to operate for the your
// creations efficiently, and "define it as a structure". I think it will be good
// training for your skill. 
