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The Frequency Domain Sinusoidal tidal waves Copy of Katsushika Hokusai The Great Wave off Kanagawa at http://commons.wikimedia.org/wiki/File:The_Great_Wave_off_Kanagawa.jpg
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Domains  Images can be represented in different domains    Spatial domain – the strength of light at points in space Frequency domain – the strength of patterns within an image Frequency domain is useful for    2 Image analysis Image compression Efficient processing
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Frequency Domain  Various frequency domains     Discrete Cosine – used in image analysis and compression Fourier – used in image analysis and processing Wavelet – used in image analysis and compression Haar and others …  Each domain defines a set of patterns from which all images can be composed  The DCT and DFT domains use sinusoidal patterns 3
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Sinusoids  A sinusoidal is characterized by the amplitude, frequency, and phase 4
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Functions into sinusoids    Any function can be represented as the sum of sinusoids Consider a one dimensional ‘square wave’. Can it be represented as the sum of ‘nonsquare waves’? 0.4 0.4 0.2 0.2 2 4 2 -0.2 -0.2 -0.4 -0.4 6 4 8 6 10 8 10
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Decomposing functions into sinusoids 1 1 0.75 0.75 0.5 0.5 0.25 0.25 2 2 4 4 6 6 8 8 10 10 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 -1 Start with a sin wave of the same frequency as the square wave. This is the “base” or “fundamental” frequency. 6
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Decomposing functions into sinusoids 1 1 0.75 0.75 0.5 0.5 0.25 0.25 2 2 4 4 6 6 8 8 10 10 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 -1 Add a 3rd “harmonic” to the fundamental frequency. The amplitude is less than that of the base and the frequency 7 is 3 times that of the base.
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Decomposing functions into sinusoids 1 1 0.75 0.75 0.5 0.5 0.25 0.25 2 2 4 4 6 6 8 8 10 10 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 -1 Add a 5th “harmonic” to the fundamental frequency. The amplitude is less than that of the base and the frequency 8 is 5 times that of the base.
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Decomposing functions into sinusoids 1 1 0.5 0.5 2 2 4 4 6 6 8 8 -0.5 -0.5 -1 -1 9 Add a 7th and 9th“harmonic” to the fundamental frequency. 10 10
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Decomposing functions into sinusoids 0.75 0.75 0.5 0.5 0.25 0.25 2 2 4 4 6 6 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 10 Adding all harmonics up to the 100th. 8 8 10 10
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Sinusoids (1D)  Consider the following sinusoidal function  f can be understood as a row profile Amplitude is determined by A Phase (shift) is given by phi Frequency is controlled by u      11 if u = 1 there is one cycle spanning the image U can be understood as the ‘number of cycles per image width’
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Discrete sinusoid  Take the continuous sinusoidal function into the discrete domain via sampling at ½ unit intervals.  One effect is to place an upper limit on the frequencies that can be captured via sampling. U must be less than N/2 in order to be recoverable by the discrete samples. This is the Shannon-Nyquist limit. Higher frequencies generate aliases  
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Aliasing example    Consider a sinusoid such that u=1 over a span of 8 units. Consider a sinusoid such that u=7 over a span of 8 units. The resulting samples are identical. The two different signals are indistinguishable after sampling and hence are aliases for each other.
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Frequency Domain  The central idea in frequency domain representation is to    Find a set of orthogonal sinusoidal patterns that can be combined to form any image Any image can be expressed as the weighted sum of these basis images. The DFT and DCT are ways of decomposing an image into the sum of sinusoidal basis images
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How to determine the amplitudes?  The DCT and DFT transformations are ways of decomposing a spatial image into sinusoidal basis images.   16 The forward DCT goes from spatial to frequency. The inverse DCT goes from frequency to spatial. The DCT and DFT are invertible – no loss of information either way
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