sampling2011

# sampling2011 - #9 ECE 253a Digital Image Processing...

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Unformatted text preview: #9 ECE 253a Digital Image Processing 10/26/11 Sampling in 2 dimensions Sampling refers to making the image discrete in its spatial coordinates. To discuss this, we need to introduce notation and define some functions: The 2-D discrete delta function is defined by: δ ( n 1 , n 2 ) = braceleftBigg 1 ( n 1 , n 2 ) = (0 , 0) else The 1-D continuous delta function can be defined by: δ ( x ) = 0 for x negationslash = 0 and lim ǫ → integraldisplay ǫ − ǫ δ ( x ) dx = 1 and the 2-D continuous delta function can be defined in terms of this 1-D function by: δ ( x, y ) = δ ( x ) δ ( y ) in which case it is separable by definition. The bed-of-nails function, also called an impulsive sheet, is comb ( x, y ; Δ x, Δ y ) = ∞ summationdisplay j = −∞ ∞ summationdisplay k = −∞ δ ( x − j Δ x, y − k Δ y ) = S ( x, y ) This is composed of an infinite array of Dirac delta functions arranged in a grid of spacing (Δ x, Δ y ). The 2-D Fourier Transform pair that we will use in this handout is: F ( u, v ) = integraldisplay ∞ −∞ integraldisplay ∞ −∞ f ( x, y ) e − i 2 π ( ux + vy ) dxdy f ( x, y ) = integraldisplay ∞ −∞ integraldisplay ∞ −∞ F ( u, v ) e + i 2 π ( ux + vy ) dudv where x and y are the spatial coordinates of the original image, and u and v are the spatial frequency coordinates of the Fourier transform of the image. The extension to 3-D is obvious. This can all be written in vector notation: 1 F ( s ) = integraldisplay ∞ −∞ f ( x ) e − i 2 π s T x d x f ( x ) = integraldisplay ∞ −∞ F ( s ) e + i 2 π s T x d s where s = ( s 1 , s 2 , . . ., s N ) and the units of the coordinate s i are the inverse of the units of the corresponding spatial coordinate x i . Important property: if a function is separable, then its Fourier transform is separable. Using ↔ to denote a F.T. pair, we have if f ( x ) = producttext i f i ( x i ) and f ( x ) ↔ F ( s ) then F ( s ) = producttext i F i ( s i ) , where f i ( x i ) ↔ F i ( s i ) Let f I ( x, y ) denote a continuous, infinite extent ideal image field representing the luminance, photographic density, or some desired parameter of a physical image. In a perfect sampling system, spatial samples of the ideal image would be obtained by multiplying by the spatial sampling function S ( x, y ): f p ( x, y ) = f I ( x, y ) S ( x, y ) = f I ( x, y ) ∞ summationdisplay j = −∞ ∞ summationdisplay k = −∞ δ ( x − j Δ x, y − k Δ y ) f p ( x, y ) = ∞ summationdisplay j = −∞ ∞ summationdisplay k = −∞ f I ( j Δ x, k Δ y ) × δ ( x − j Δ x, y − k Δ y ) where it is observed that f I ( x, y ) may be brought inside the summation and evaluated only at the sample points ( j Δ x, k Δ y )....
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sampling2011 - #9 ECE 253a Digital Image Processing...

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