EDIT: There is a great introduction to the concepts here.
There's a fair bit of math behind that question. In simple terms, consider a 1-D function, such as an audio clip. The fourier transform identifies the frequencies present in that signal. Each sample in the original audio clip correlates to the amplitude of the sound wave at any given point in time. In contrast, each sample in the fourier transform identifies the amplitude of a particular frequency of oscillation. For example, a pure sine wave at 1 kHz will have a fourier transform with a single spike at the 1 kHz mark. Audio waves are combinations of many different sine waves, and the fourier transform isolates which sine waves are contributing and by how much. (Note that the real explanation requires delving into complex numbers, but the foregoing gives the essence of what's going on).
The fourier transform of an image is a simple extension of the 1-D fourier transform into two dimensions, and is achieved by simply applying the 1-D transform to each row of an image, and then transforming each column of the resulting image. It produces essentially the same thing. A picture of smooth water waves travelling in a diagonal direction will transform to a series of spikes along that same diagonal.
The fourier transform is defined over continuous functions. The FFT is an technique for efficiently evaluating the fourier transform over discrete sets of data.
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编辑：此处对概念有很好的介绍。
这个问题背后有相当多的数学知识。简而言之，考虑一维函数，例如音频剪辑。傅里叶变换识别该信号中存在的频率。原始音频剪辑中的每个样本都与任何给定时间点的声波振幅相关。相反，傅里叶变换中的每个样本都识别特定振荡频率的幅度。例如，1 kHz 的纯正弦波将进行傅立叶变换，并在 1 kHz 标记处有一个尖峰。音频波是许多不同正弦波的组合，傅立叶变换隔离了哪些正弦波做出了贡献以及贡献了多少。 （请注意，真正的解释需要深入研究复数，但前面给出了正在发生的事情的本质）。
图像的傅立叶变换是一维傅立叶变换到二维的简单扩展，通过简单地将一维变换应用于图像的每一行，然后变换所得图像的每一列来实现。它产生本质上相同的东西。沿对角线方向传播的平滑水波的图片将转变为沿同一对角线的一系列尖峰。
傅立叶变换是在连续函数上定义的。 FFT 是一种有效评估离散数据集傅立叶变换的技术。
EDIT: There is a great introduction to the concepts here.
There's a fair bit of math behind that question. In simple terms, consider a 1-D function, such as an audio clip. The fourier transform identifies the frequencies present in that signal. Each sample in the original audio clip correlates to the amplitude of the sound wave at any given point in time. In contrast, each sample in the fourier transform identifies the amplitude of a particular frequency of oscillation. For example, a pure sine wave at 1 kHz will have a fourier transform with a single spike at the 1 kHz mark. Audio waves are combinations of many different sine waves, and the fourier transform isolates which sine waves are contributing and by how much. (Note that the real explanation requires delving into complex numbers, but the foregoing gives the essence of what's going on).
The fourier transform of an image is a simple extension of the 1-D fourier transform into two dimensions, and is achieved by simply applying the 1-D transform to each row of an image, and then transforming each column of the resulting image. It produces essentially the same thing. A picture of smooth water waves travelling in a diagonal direction will transform to a series of spikes along that same diagonal.
The fourier transform is defined over continuous functions. The FFT is an technique for efficiently evaluating the fourier transform over discrete sets of data.
Mathworks 的 Steve Eddins 已经在他的博客上讨论傅里叶变换有一段时间了 - 您应该请查看此处。
Steve Eddins of Mathworks has been discussing Fourier Transforms in general on his blog for a while now - you should check it out here.