This is more compactly expressed in vector notation, where we define formula_211 and formula_212 as "d"-dimensional vectors of indices from 0 to formula_213, which we define as formula_214:  where the division formula_216 is defined as formula_217 to be performed element-wise, and the sum denotes the set of nested summations above. The inverse of the multi-dimensional DFT is, analogous to the one-dimensional case, given by:  As the one-dimensional DFT expresses the input formula_42 as a superposition of sinusoids, the multidimensional DFT expresses the input as a superposition of plane waves, or multidimensional sinusoids. The direction of oscillation in space is formula_220. The amplitudes are formula_221. This decomposition is of great importance for everything from digital image processing (two-dimensional) to solving partial differential equations. The solution is broken up into plane waves. The multidimensional DFT can be computed by the composition of a sequence of one-dimensional DFTs along each dimension. In the two-dimensional case formula_222 the formula_223 independent DFTs of the rows (i.e., along formula_224) are computed first to form a new array formula_225. Then the formula_226 independent DFTs of "y" along the columns (along formula_227) are computed to form the final result formula_228. Alternatively the columns can be computed first and then the rows. The order is immaterial because the nested summations above commute. An algorithm to compute a one-dimensional DFT is thus sufficient to efficiently compute a multidimensional DFT. This approach is known as the "row-column" algorithm. There are also intrinsically multidimensional FFT algorithms. For input data formula_204 consisting of real numbers, the DFT outputs have a conjugate symmetry similar to the one-dimensional case above:  where the star again denotes complex conjugation and the formula_206-th subscript is again interpreted modulo formula_232 (for formula_233). The DFT has seen wide usage across a large number of fields; we only sketch a few examples below (see also the references at the end). All applications of the DFT depend crucially on the availability of a fast algorithm to compute discrete Fourier transforms and their inverses, a fast Fourier transform. When the DFT is used for signal spectral analysis, the formula_234 sequence usually represents a finite set of uniformly spaced time-samples of some signal formula_235, where formula_236 represents time.