Chapter 4: Image Filtering - Engineeringmtoledo/6088/2004/handout_ch4.pdf · • Convolution in the...
Transcript of Chapter 4: Image Filtering - Engineeringmtoledo/6088/2004/handout_ch4.pdf · • Convolution in the...
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Chapter 4: Image Filtering
INEL 6088 Lecture Notes – Computer Vision
Prof. M. Toledo - Fall 2004
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Histogram Modification (to improve contrast)
Scaling: maps pixels in the range [a,b] to [z1,zk].
A pixel value z will have a new pixel value z’:
11)( zzzz kabaz +−=′ −
−
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If the desired gray value distribution is known a priori:
Original: pi pixels at ziDesired: qi pixels at zi
Start from left select k1so
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Then find k2 such that
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Systems Formalism
Kernel or mask
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Characteristics of Convolution
• Linear (convolution o a sum is the sum of the convolutions)
• Convolution of a scaled image is the scaled convolution
• Spatially invariant
• Convolution in the image domain correspond to multiplication in the (spatial) frequency domain
• Frequency-domain convolution used only for very large kernels – not common in machine vision
• The kernel defines the filter being used
• We use non-linear filters as well; formalism does not apply but many are used in a very similar way.
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Noise-free
Salt & pepper -Random black & white pixels
Impulse -Only random white
Gaussian –random gray level variations
Common Types of Noise
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Mean Filter
Smoothing filter: pixel weights are not equal, but are usually symmetric and add 1.
Single-peak, or main lope
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3×3 5×5 7×7
Mean Filter Applied to Noisy Images
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Result of using the smoothing filter seen previously
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Median Filter
• Nonlinear• Very effective to remove salt & pepper and impulse!• Retain image detail
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Results of Applying a Median Filter on Previous Noisy Images
3×3 5×5 7×79/21/2004 INEL 6088 Lecture Notes - Ch. 4 -
Image Filtering12
Gaussian Smoothing
22
22
],[ σji
ejig+−
=
3
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Properties of Gaussian Filter
• Rotational symmetry• Single lope in both space and frequency
domain• Smoothing is controlled by a single
parameter, �
• Separable• Will spread impulse and salt & pepper
noise!
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Gaussian Mean
space
frequency
Multiple lopes can cause halos on image
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• To generate the 1D kernel, we must discretize the gaussian• Require the mask width to subtend most of the gaussian area:
use w = 5�
for 98.76% of the area; for a
�
=0.6, use w = 3 pixels; for
�
=1, use w = 5
Gaussian filter – can be applied as the convolution of two 1D filters
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• Use discrete kernel entries to reduce filtering times – normalize the smallest kernel entry to make it 1
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Fig 4.13 – Result of using a single horizontal convolution mask k on (a) original noisy image. (b) After convolution with k. (c) transpose of (b). (d) convolution of (c) with k. (e) Transpose of (d) to get the final smoothed image.
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Design of Gaussian Filters• Use the row n of Pascal’s Triangle as a one-
dimensional, n-point approximation of a Gaussian filter.
1615151561
15101051
14641
1331
121
11
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Fig. 4-14: Using the fifth row of Pascal’s triangle as a Gaussian filter. (a) original; (b) After smoothing in the horizontal dir. (c) After smoothing in the vertical direction
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Another approach: compute the mask directly from the discrete Gaussian distribution. Below it is shown the result of using
2
22
2],[ σji
cjig e
+−=
For n=7 and �2=2.
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After multiplying by 91 so that the smallest values (corners) be 1,
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Fig. 4-15: 3-D plot of the 7x7 Gaussian mask.
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The result of the convolution must be divided by the sum of the mask weights (1115) to ensure that regions of uniform intensity are not affected.
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Fig. 4-16: Result of Gaussian smoothing using the 7x7 mask.
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