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Pendahuluan,[object Object],BambangHeruIswanto, Dr.rer.nat,[object Object],Sesion #01,[object Object],JurusanFisika,[object Object],FakultasMatematikadanIlmuPengetahuanAlam,[object Object]
Outline ,[object Object],Transforms,[object Object],Fourier Transform,[object Object],Time Domain and Frequency Domain,[object Object],Discrete Fourier Transform,[object Object],Fast Fourier Transform,[object Object],Applications,[object Object],Applications: Frequency Domain In Images,[object Object],Other Applications of the DFT,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object],2,[object Object],04/01/2011,[object Object]
04/01/2011,[object Object],3,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Transforms,[object Object],Transform:,[object Object],In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica),[object Object],Can be thought of as a substitution,[object Object],04/01/2011,[object Object],4,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Transforms,[object Object],Example of a substitution:,[object Object],Original equation: x  + 4x² – 8 = 0,[object Object],Familiar form: ax² + bx + c = 0,[object Object],Let: y = x²,[object Object],Solve for y,[object Object],x = ±√y,[object Object],4,[object Object],04/01/2011,[object Object],5,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Fourier Transform,[object Object],Property of transforms:,[object Object],They convert a function from one domain to another with no loss of information,[object Object],Fourier Transform:,[object Object],   converts a function from the time (or spatial) domain to the frequency domain,[object Object],04/01/2011,[object Object],6,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Time Domain and Frequency Domain,[object Object],Time Domain:,[object Object],Tells us how properties (air pressure in a sound function, for example) change over time:,[object Object],Amplitude = 100,[object Object],Frequency = number of cycles in one second = 200 Hz,[object Object],04/01/2011,[object Object],7,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Time Domain and Frequency Domain,[object Object],Frequency domain:,[object Object],Tells us how properties (amplitudes) change over frequencies:,[object Object],04/01/2011,[object Object],8,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Time Domain and Frequency Domain,[object Object],Example:,[object Object],Human ears do not hear wave-like oscilations, but constant tone,[object Object],Often it is easier to work in the frequency domain,[object Object],04/01/2011,[object Object],9,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Time Domain and Frequency Domain,[object Object],In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions,[object Object],In picture: the composition of the first two functions gives the bottom one,[object Object],04/01/2011,[object Object],10,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Time Domain and Frequency Domain,[object Object],04/01/2011,[object Object],11,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Fourier Transform,[object Object],Because of the ,[object Object],    property:,[object Object],Fourier Transform takes us to the frequency domain:,[object Object],04/01/2011,[object Object],12,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Discrete Fourier Transform,[object Object],In practice, we often deal with discrete functions (digital signals, for example),[object Object],Discrete version of the Fourier Transform is much more useful in computer science:,[object Object],O(n²) time complexity,[object Object],04/01/2011,[object Object],13,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Fast Fourier Transform,[object Object],Many techniques introduced that reduce computing time to O(n log n),[object Object],Most popular one: radix-2 decimation-in-time (DIT) FFT Cooley-Tukey algorithm:,[object Object],                     (Divide and conquer),[object Object],04/01/2011,[object Object],14,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Applications,[object Object],In image processing:,[object Object],Instead of time domain: spatial domain (normal image space),[object Object],frequency domain: space in which each image value at image position F represents the amount that the intensity values in image I vary over a specific distance related to F ,[object Object],04/01/2011,[object Object],15,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Applications: Frequency Domain In Images,[object Object],If there is value 20 at the point that represents the frequency 0.1 (or 1 period every 10 pixels). This means that in the corresponding spatial domain image I the intensity values vary from dark to light and back to dark over a distance of 10 pixels, and that the contrast between the lightest and darkest is 40 gray levels ,[object Object],04/01/2011,[object Object],16,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Applications: Frequency Domain In Images,[object Object],Spatial frequency of an image refers to the rate at which the pixel intensities change,[object Object],In picture on right:,[object Object],High frequences:,[object Object],Near center,[object Object],Low frequences:,[object Object],Corners,[object Object],04/01/2011,[object Object],17,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Other Applications of the DFT,[object Object],Signal analysis,[object Object],Sound filtering,[object Object],Data compression,[object Object],Partial differential equations,[object Object],Multiplication of large integers,[object Object],04/01/2011,[object Object],18,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]
Percobaan DFT mengunakan Labview,[object Object],04/01/2011,[object Object],19,[object Object],©  2010 Universitas Negeri Jakarta   |  www.unj.ac.id                      |,[object Object]

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