二维卷积定理的证明

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卷积定理

若$$\begin{gathered} g\ast h={\iint }_{-\infty }^{\infty }g(\alpha ,\beta )h(x-\alpha ,y-\beta )d\alpha d\beta \\ F[g(x,y)]={\iint }_{-\infty }^{\infty }g(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]dxdy=G(f_x,f_y) \\ F[h(x,y)]={\iint }_{-\infty }^{\infty }h(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]dxdy=H(f_x,f_y) \end{gathered}$$
则可得到
$$\begin{gathered} F[g\ast h]=G\cdot H \\ F[g\cdot h]=G\ast H \end{gathered}$$

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一、 $F[g\ast h]=G\cdot H$ 的证明

$$\begin{array}{c}\begin{array}{rl}F[g\ast h]& ={\iint }_{-\infty }^{\infty }g(x,y)\ast h(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]dxdy\\ & ={\iint }_{-\infty }^{\infty }{\iint }_{-\infty }^{\infty }g(\alpha ,\beta )h(x-\alpha ,y-\beta )d\alpha d\beta \cdot exp[-i2\pi (f_{x}x+f_{y}y)]dxdy\\ & ={\iint }_{-\infty }^{\infty }{\iint }_{-\infty }^{\infty }g(\alpha ,\beta )h(x-\alpha ,y-\beta )d\alpha d\beta \cdot exp[-i2\pi (f_x(x-\alpha )+f_y(y-\beta ))]\cdot exp[-i2\pi (f_x\alpha +f_y\beta )]dxdy\\ & ={\iint }_{-\infty }^{\infty }g(\alpha ,\beta ){\iint }_{-\infty }^{\infty }h(x-\alpha ,y-\beta )exp[-i2\pi (f_x(x-\alpha )+f_y(y-\beta ))]dxdy\cdot exp[-i2\pi (f_x\alpha +f_y\beta )]d\alpha d\beta \\ & ={\iint }_{-\infty }^{\infty }g(\alpha ,\beta )H(f_x,f_y)\cdot exp[-i2\pi (f_x\alpha +f_y\beta )]d\alpha d\beta \\ & =G(f_x,f_y)\cdot H(f_x,f_y)\end{array}\end{array}$$

二、 $F[g\cdot h]=G\ast H$ 的证明

$$\begin{array}{c}\begin{array}{rl}F[g\cdot h]& ={\iint }_{-\infty }^{\infty }g(x,y)\cdot h(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]dxdy\\ & ={\iint }_{-\infty }^{\infty }{\iint }_{-\infty }^{\infty }G(f_x,f_y)exp[i2\pi (f_{x}x+f_{y}y)]d{f}_{x}d{f}_y\cdot h(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]dxdy\\ & ={\iint }_{-\infty }^{\infty }{\iint }_{-\infty }^{\infty }G(\alpha ,\beta )exp[i2\pi (\alpha x+\beta y)]d\alpha d\beta \cdot h(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]dxdy\\ & ={\iint }_{-\infty }^{\infty }G(\alpha ,\beta ){\iint }_{-\infty }^{\infty }h(x,y)exp[-i2\pi (f_{x}x+f_{y}y)]exp[i2\pi (\alpha x+\beta y)]dxdyd\alpha d\beta \\ & ={\iint }_{-\infty }^{\infty }G(\alpha ,\beta ){\iint }_{-\infty }^{\infty }h(x,y)exp[-i2\pi ((f_x-\alpha )x+(f_y-\beta )y)]dxdyd\alpha d\beta \\ & ={\iint }_{-\infty }^{\infty }G(\alpha ,\beta )H(f_x-\alpha ,f_y-\beta )d\alpha d\beta \\ & =G(f_x,f_y)\ast H(f_x,f_y)\end{array}\end{array}$$

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总结

第一个公式的证明利用了自变量$x,y$的变换，第二个公式的证明利用了逆傅里叶变换和相移定理。