Spatial operations

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? Spatial operations
- Smoothing Smoothing filters are used for
blurring and for noise reduction. Blurring is
used in preprocessing steps, such as removal of
small details from an image. Spatial averaging
Each pixel is replaced with a weighted average of
its neighborhood pixels, that is
where y(m,n) and v(m,n) are the input and output
images, respectively, W is a suitably chosen
window, and a(k,l) is an impulse response called
spatial mask.
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A common class of spatial averaging filters has
all equal weights, giving
where a(k,l)1/NW and NW is the number of pixels
in the window.
some spatial averaging masks
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Median Filtering Each pixel is replaced by the
median of the pixels contained in a window W
around the center pixel, that is
It is classified into a non-linear filter, such
as order statistics. Some order statistic filters
use the minimum or maximum value in a given
window. It reduces impulsive nose and preserves
edges well. Caution The results between 1D
separable median and 2D normal median filters are
not unique because of non-linearity. Ex.) Let
y(m)2,3,8,4,2 and W-1,0,1. Assume the
values of the outsides of the sequence are zero.
v(0) median 0,2,3 2, v(1) median
2,3,8 3 v(2) median 3,8,4 4, v(3)
median 8,4,2 4 v(4) median 4,2,0 2
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Hybrid Filtering non-linear(order statistics)
and linear(spatial averaging) filters are
combined.
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comparison between spatial averaging and median
filter.
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- Sharpening(Crispening) Psychophysical
experiments indicate that a photograph or visual
signal with accentuated or crispened edges is
often more subjectively pleasing than exact
photometric reproduction. Unsharp Masking The
unsharp masking technique is used commonly in the
printing industry for crispening the edges. A
signal proportional to the unsharp, or low-pass
filtered, version of the image is subtracted form
the image. This is equivalent to adding the
gradient, or high-passed signal, to the image.
The unsharp masking operation can be represented
by
where HLP and HHP mean low-pass and high-pass
filtering, respectively. The constant (a-1) is
typically chosen as 0.25-0.33.
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Statistical Differencing The stastistical
differencing, suggested by Wallis, forces the
enhanced image to a form with desired mean and
standard deviation. The operation is defined by
where ?(m,n) and m(m,n) represent local mean and
standard deviation, ?d and md denote desired mean
and standard deviation, ? is a gain factor that
prevents overly large output values when ?(m,n)
is small, and ? is a factor controlling the ratio
of the edge to background intensities. The
constant ?d, md, ? and ? are typically chosen as
8.5, 128, 1/6 and 0.1.
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- Homomorphic filtering Homomorphic filtering is
a useful technique for image enhancement when an
image is subject to multiplicative noise or
interference and for image deconvolution. It can
be performed in both spatial and frequency
domains. One can reduce the dynamic range and
increase the local contrast of an image to be
enhanced by applying a homomorphic filtering to
an illumination-reflectance image model. Based on
the image model, an input image u(m,n) can be
expressed as
where i(m,n) represents the illumination and
r(m,n) represents the reflectance. The
illumination i(m,n) is assumed to be the primary
contributor to the dynamic range and is assumed
to vary slowly, while the reflectance r(m,n) that
represents the detail of an object is assumed to
be the primary contributor to local contrast and
is assumed to vary rapidly.
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To separate i(m,n) from r(m,n), a logarithm
operation is applied to the previous equation,
and the result is
If we assume that log i(m,n) remains slowly
varying and log r(m,n) remains rapidly varying,
lowpass filtering log u(m,n) will result in log
i(m,n) and highpass filtering log u(m,n) will
result in log r(m,n).
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- Zooming(Interpolation) Various interpolation
techniques can be used in changing the size of a
digital image to improve its appearance when
viewed on a display device. Digital zooming can
be performed by using a continuous interpolator
which reconstructs a continuous signal from
samples.
where hc(x) denotes a continuous interpolation
function.
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1) Sinc-function interpolator
This interpolation function is not spatially
limited.
2) Nth order interpolator - zero-order
interpolator
- first-order interpolator
- second-order interpolator
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- cubic spline(third-order) interpolator
2D continuous interpolator
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- L1 decimator L1 decimator reduces sampling
rate by L.
The decimated sequence v(m) can be written as
The LPF guarantees aliasing effects. Caution
Decimations are different from sub-sampling.
Sub-sampling doesnt have such a LPF. It only
reduces the sampling rate.
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? Transform operation
- Linear filtering Linear filtering in frequency
domain is straightforward. We simply compute the
Fourier transform U(k,l) of the image to be
enhanced, and multiply the result by a filter
transfer function H(k,l).
Then we obtained the enhanced image by taking the
inverse Fourier transform of V(k,l).
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? ????

• Anil K. Jane, Fundamentals of Digital Image
  Processing, Prentice Hall International Edition,
  Singapore, ch. 7, 1989.
• Rafael C. Gonzalez and Richard E. Woods, Digital
  Image Processing, Addison Wesley, Massachusetts,
  ch. 4, 1992.
• Jae S. Lim, Two-Dimensional Signal and Image
  Processing, Prentice Hall International Edition,
  New Jersey, ch. 8, 1990.