Digital Image Processing

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Digital Image Processing

• Image Enhancement
• Spatial Filtering

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Image Enhancement Spatial Filtering

• Image enhancement in the spatial domain can be
  represented as
• g(m,n) T(f)(m,n)
• The transformation T maybe linear or nonlinear.
  We will mainly study linear operators T but
  will see one important nonlinear operation.

Transformation
Enhanced Image
Given Image
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How to specify T

• If the operator T is linear and shift
  invariant (LSI), characterized by the
  point-spread sequence (PSS) h(m,n) , then
  (recall convolution)

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How to specify T

• In practice, to reduce computations, h(m,n)
  is of finite extent
• h(k,l) 0 for (k,l) ? ?
• where D is a small set (called neighborhood). D
  is also called as the support of h.
• In the frequency domain, this can be represented
  as
• G(u,v) He(u,v) Fe(u,v)
• where He(u,v) and Fe(u,v) are obtained after
  appropriate zeropadding.

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How to specify T

• Many LSI operations can be interpreted in
  the frequency domain as a filtering operation.
  It has the effect of filtering frequency
  components (passing certain frequency components
  and stopping others).
• The term filtering is generally associated
  with such operations.

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How to specify T

• Examples of some common filters (1-D case)

Lowpass filter
Highpass filter
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• If h(m, n) is a 3 by 3 mask given by
• then

w1 w2 w3
h w4 w5 w6
w7 w8 w9
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• The output g(m, n) is computed by sliding the
  mask over each pixel of the image f(m, n).
  This filtering procedure is sometimes referred
  to as moving average filter.
• Special care is required for the pixels at
  the border of image f(m, n). This depends on
  the so-called boundary condition. Common
  choices are
• The mask is truncated at the border (free
  boundary)
• The image is extended by appending extra
  rows/columns at the boundaries. The extension
  is done by repeating the first/last
  row/column or by setting them to some
  constant (fixed boundary).
• The boundaries wrap around (periodic boundary).

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• In any case, the final output g(m, n) is
  restricted to the support of the original image
  f(m, n).
• The mask operation can be implemented in MATLAB
  using the filter2 command, which is based
  on the conv2 command.

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Smoothing Filters

• Image smoothing refers to any image-to-image
  transformation designed to smooth or
  flatten the image by reducing the rapid
  pixel-to-pixel variation in grayvalues.
• Smoothing filters are used for
• Blurring This is usually a preprocessing step
  for removing small (unwanted) details before
  extracting the relevant (large) object,
  bridging gaps in lines/curves,
• Noise reduction Mitigate the effect of noise
  by linear or nonlinear operations.

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Image smoothing by averaging (lowpass
spatial filtering)

• Smoothing is accomplished by applying an
  averaging mask.
• An averaging mask is a mask with positive
  weights, which sum to 1. It computes a weighted
  average of the pixel values in a neighborhood.
  This operation is sometimes called
  neighborhood averaging.
• Some 3 x 3 averaging masks
• This operation is equivalent to lowpass
  filtering.

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Example of Image Blurring
Original Image
Avg. Mask
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Example of Image Blurring
N 3
N 7
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Example of Image Blurring
N 11
N 21
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Example of noise reduction
Noise-free Image
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Example of noise reduction
Zero-mean Gaussian noise, Variance 0.01
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Example of noise reduction
Zero-mean Gaussian noise, Variance 0.05
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Median Filtering

• The averaging filter is best suited for noise
  whose distribution is Gaussian
• The averaging filter typically blurs edges and
  sharp details.
• The median filter usually does a better job of
  preserving edges.
• Median filter is particularly suited if the noise
  pattern exhibits strong (positive and negative)
  spikes. Example salt and pepper noise.

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Median Filtering

• Median filter is a nonlinear filter, that also
  uses a mask. Each pixel is replaced by the median
  of the pixel values in a neighborhood of the
  given pixel.
• Suppose A a1, a2, , ak are the pixel values
  in a neighborhood of a given pixel with a1 ? a2 ?
  ? ak. Then
• Note Median of a set of values is the center
  value, after sorting.
• For example If A 0,1,2,4,6,6,10,12,15 then
  median(A) 6.

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Example of noise reduction
Gaussian noise s 0.2 Salt Pepper noise
prob. 0.2
MSE 0.0337
MSE 0.062
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Example of noise reduction
Output of 3x3 Averaging filter
MSE 0.0075
MSE 0.0125
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Example of noise reduction
Output of 3x3 Median filter
MSE 0.0042
MSE 0.0089
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Image Sharpening

• This involves highlighting fine details or
  enhancing details that have been blurred.

Before
After Sharpening
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Basic highpass spatial filtering

• This can be accomplished by a linear
  shift-invariant operator, implemented by means of
  a mask, with positive and negative coefficients.
• This is called a sharpening mask, since it tends
  to enhance abrupt graylevel changes in the image.
• The mask should have a positive coefficient at
  the center and negative coefficients at the
  periphery. The coefficients should sum to zero.

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Basic highpass spatial filtering

• Example
• This is equivalent to highpass filtering.
• A highpass filtered image g can be thought of as
  the difference between the original image f and a
  lowpass filtered version of f
• g(m,n) f(m,n) lowpass(f(m,n))

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Example
Original Image
Highpass filtering
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High-boost filtering

• This is a filter whose output g is produced by
  subtracting a lowpass (blurred) version of f from
  an amplified version of f
• g(m,n) Af(m,n) lowpass(f(m,n))
• This is also referred to as unsharp masking.

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High-boost filtering

• Observe that
• g(m,n) Af(m,n) lowpass(f(m,n))
• (A-1)f(m,n) f(m,n) lowpass(f(m,n))
• (A-1)f(m,n) hipass(f(m,n))
• For 1 gt A , part of the original image is added
  back to the highpass filtered version of f.
• The result is the original image with the edges
  enhanced relative to the original image.

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Example
Original Image
Highpass filtering
High-boost filtering
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Derivative filter

• Averaging tends to blur details in an image.
  Averaging involves summation or integration.
• Naturally, differentiation or differencing
  would tend to enhance abrupt changes, i.e.,
  sharpen edges.
• Most common differentiation operator is the
  gradient

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Derivative filter

• The magnitude of the gradient is
• Discrete approximations to the magnitude of the
  gradient is normally used.

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Derivative filter

• Consider the following image region
• We may use the approximation

z1 z2 z3
z4 z5 z6
z7 z8 z9
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Derivative filter

• This can implemented using the masks
• As follows

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Derivative filter

• Alternatively, we may use the approximation
• This can implemented using the masks
• As follows

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Derivative filter

• The resulting maks are called Roberts
  cross-gradient operators.
• The Roberts operators and the Prewitt/Sobel
  operators (described later) are used for edge
  detection and are sometimes called edge detectors.

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Example Roberts cross-gradient operator
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Example Roberts cross-gradient operator
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Prewitt operators

• Better approximations to the gradient can be
  obtained by
• This can be implemented using the masks
• as follows

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Sobel operators

• Another approximation is given by the masks
• The resulting masks are called Sobel operators.

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Example
Prewitt
Sobel