The document discusses digital image processing and noise reduction techniques. It covers: - The sources and types of noise that can arise in digital images during acquisition and transmission. - Common noise models including additive Gaussian noise, salt and pepper noise, and speckle noise. - Spatial and frequency domain filtering methods for noise reduction, such as mean filters, median filters, and Wiener filtering. - Wavelet domain techniques like wavelet thresholding that can effectively remove noise while preserving image details.

1. P. Nirmala devi
AP(SLG)/ECE
KEC

2.  Digital Image Processing is the use of computer
algorithms to perform image processing on digital
images.
 Advantages over analog image processing:
- Allows a much wider range of
algorithms to be applied to the input data
- Avoid problems such as the build-up of
noise and signal distortion during
processing.
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3.  Two principal application areas:
◦ Improvement of pictorial information for human
interpretation
◦ Processing of image data for storage, transmission and
representation for autonomous machine perception
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4. ◦ An digital image may be defined as a
two-dimensional quantity f(x,y)
x and y are spatial coordinates and
f is intensity or gray level at that point
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5.  Low-level IP
Image preprocessing to reduce noise, contrast
enhancement, image sharpening
Both inputs and outputs are images
 Mid-level IP
Segmentation and description
The inputs are generally images, but outputs are
attributes extracted from those images
(e.g., edges,contours… )
 High-level IP
Making sense of an ensemble of recognized objects
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6.  Goal:
◦ To Remove noise
◦ To Preserve useful information
 Applications:
◦ Medical signal/image analysis (ECG, CT, MRI etc.)
◦ Data mining
◦ Radio astronomy image analysis
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7. Visually unpleasant
Bad for compression
Bad for analysis
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9.  Images are often contaminated by noise during
i) acquisition
ii) storage
iii)transmission
 Effect:
Degradation at the quality of the images
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10. The sources of noise in digital images arise during
image acquisition (digitization) and transmission
◦ Imaging sensors can be affected by ambient
conditions
◦ Interference can be added to an image during
transmission
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thermal imaging electrical interference
ultrasound imaging
physical interference

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 Simplified assumptions
 Noise is independent of signal
 Noise types
 Independent of spatial location

Impulse noise

Additive white Gaussian noise
 Spatially dependent

Periodic noise

13.  Definition: is considered to be any
measurement that is not part of the phenomena
of interest.
 Images are affected by different types of noise:
 Gaussian noise
 Salt and Pepper noise
 Poisson Noise
 Speckle Noise
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14.  Impulse noise
Characterized by some portion of image pixels
that are corrupted, leaving the remaining pixels
unchanged. (Salt & Pepper Noise)
 Additive noise
A value from a certain distribution is added to each
image pixel, for example, a Gaussian distribution.
 Multiplicative noise
The intensity of the noise varies with the signal
intensity (e.g., speckle noise).
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15. W
j
H
i
j
i
X
j
i
Y
≤
≤
≤
≤





=
1
,
1
)
,
(
0
255
)
,
(
Definition
Each pixel in an image has the probability of p/2 (0<p<1) being
contaminated by either a white dot (salt) or a black dot (pepper)
with probability of p/2
with probability of p/2
with probability of 1-p
noisy pixels
clean pixels
X: noise-free image, Y: noisy image
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128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 255 0 128 128 128 128 128 128
128 128 128 128 0 128 128 128 128 0
128 128 128 128 128 128 128 128 128 128
128 128 0 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128 128 128
0 128 128 128 128 255 128 128 128 128
128 128 128 128 128 128 128 128 128 255
128 128 128 128 128 128 128 255 128 128

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Noisy image Y
filtering
algorithm
Can we make the denoised image X as close
to the noise-free image X as possible?
^
denoised
image

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225 225 225 226 226 226 226 226
225 225 255 226 226 226 225 226
226 226 225 226 0 226 226 255
255 226 225 0 226 226 226 226
225 255 0 225 226 226 226 255
255 225 224 226 226 0 225 226
226 225 225 226 255 226 226 228
226 226 225 226 226 226 226 226
0 225 225 226 226 226 226 226
225 225 226 226 226 226 226 226
225 226 226 226 226 226 226 226
226 226 225 225 226 226 226 226
225 225 225 225 226 226 226 226
225 225 225 226 226 226 226 226
225 225 225 226 226 226 226 226
226 226 226 226 226 226 226 226
Sorted: [0, 0, 0, 225, 225, 225, 226, 226, 226]

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Noisy image Y
denoised
image
3-by-3 window

20. W
j
H
i
N
j
i
N
j
i
N
j
i
X
j
i
Y
≤
≤
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1
,
1
),
,
0
(
~
)
,
(
),
,
(
)
,
(
)
,
(
2
σ
Definition
Each pixel in an image is disturbed by a Gaussian random variable
With zero mean and variance σ2
X: noise-free image, Y: noisy image
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Numerical Example
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 128 128 128 128 128 128
128 128 129 127 129 126 126 128
126 128 128 129 129 128 128 127
128 128 128 129 129 127 127 128
128 129 127 126 129 129 129 128
127 127 128 127 129 127 129 128
129 130 127 129 127 129 130 128
129 128 129 128 128 128 129 129
128 128 130 129 128 127 127 126

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 A different type of noise in the coherent imaging of
objects caused by errors in data transmission
 Speckle noise follows a gamma distribution
 Presence of speckle is undesirable
 damages radiometric resolution
 affects the tasks of human interpretation and scene analysis.

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Image Denoising Techniques
 Spatial Domain Denoising
• Conventional AND Adaptive filtering
•Frequency Domain Denoising
• Wiener Filtering
•Wavelet Domain Denoising
• Wavelet thresholding: Hard vs. Soft
• Wavelet-domain shrinking
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24.  Spatial filters are designed to highlight or
suppress specific features in an image, based on
their spatial frequency.
 Linear Filters - Mean Filters
 Non Linear Filters - Median Filters
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25. A common filtering involves moving a 'window' of a few pixels in dimension
(e.g. 3x3, 5x5, etc.) over each pixel in the image, applying a mathematical
calculation using the pixel values under that window, and replacing the central
pixel with the new value.
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26. Image of CHURN Farm
Daedalus 1268 ATM
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27. A low-pass filter is designed to emphasise larger, homogeneous areas of
similar tone and reduce the smaller detail in an image. Thus, low-pass filters
generally serve to smooth the appearance of an image.
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28. A high-pass filter does the opposite, and serves to sharpen the appearance of fine
detail in an image.
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33. Wide windows do not provide good localization
at high frequencies.
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34. Use narrower windows at high frequencies.
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35. Narrow windows do not provide good localization
at low frequencies.
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36. Use wider windows at low frequencies.
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39.  Overcomes the preset resolution problem of the
STFT by using a variable length window:
◦ Use narrower windows at high frequencies for better
time resolution.
◦ Use wider windows at low frequencies for better
frequency resolution.
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45.  There are many different wavelets:
Morlet
Haar Daubechies
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46.  Sparsity: for functions typically found in practice,
many of the coefficients in a wavelet
representation are either zero or very small.
 Linear-time complexity: many wavelet
transformations can be accomplished in O(N)
time.
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47. • Adaptability: wavelets can be adapted to
represent a wide variety of functions (e.g.,
functions with discontinuities, functions defined on
bounded domains etc.).
– Well suited to problems involving images, open or
closed curves, and surfaces of just about any variety.
– Can represent functions with discontinuities or corners
more efficiently (i.e., some have sharp corners
themselves).
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Properties of Wavelets (cont’d)
•Multiresolution analysis :
•Multiresolution analysis: representation of a
signal (e.g., an images) in more than one
resolution/scale.
•Features that might go undetected at one
resolution may be easy to spot in another.

49. • Noise filtering
• Image compression
– Fingerprint compression
• Image fusion
• Recognition
• Image Matching and Retrieval
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50. One Stage Filtering gives Approximations and
details:
• The low-frequency content is the most
important part in many applications, and
gives the signal its identity.
This part is called “Approximations”
• The high-frequency gives the ‘flavor’, and
is called “Details”
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51.  Perceptually flat regions should be flat
 Image boundaries should be preserved (neither
blurred or sharpened)
 Texture details should not be lost
 Global contrast should be preserved
 No artifacts should be generated
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52.  Different sources and type of noises
 How strong is the noise?
 Locally, it is hard to distinguish
◦ Texture vs. noise
◦ Object boundary vs. structural noise
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53.  We need to distinguish spatially-localized events
(edges) from noise components
 Wavelet denoising attempts to remove the noise
present in the signal while preserving the signal
characteristics, regardless of its frequency
content.
 What are essential features of the data, and what
features are “noise”?
 To know more about noise components
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Forward
Transform
Inverse
Transform
Denoising
operation
e.g.,
KLT
DCT
WT
e.g.,
Linear Wiener filtering
Nonlinear Thresholding
Noisy
signal
denoised
signal

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56.  DWT of the image is calculated
 Resultant coefficients are passed through
threshold testing
 The coefficients < threshold are removed, others
shrinked
 Resultant coefficients are used for image
reconstruction with IWT.
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57.  Methods Used
◦ Universal Thresholding
◦ Visu Shrink
◦ Sure Shrink
◦ Bayes Shrink
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58.  Wavelet thresholding (first proposed by Donoho) is a
signal estimation technique that exploits the capabilities
of wavelet transform for signal denoising.
 It removes noise by killing coefficients that are
insignificant relative to some threshold.
 Types
◦ Universal or Global Thresholding
 Hard
 Soft
◦ SubBand Adaptive Thresholding
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59. 
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
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<
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
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otherwise
T
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DWT IWT
Thresholding
Y X
~
Soft thresholding
Hard thresholding
Noisy
signal
denoised
signal
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 The hard thresholding operator is
defined as
D(U, λ) = U for all |U|> λ
 Hard threshold is a “keep or kill”
procedure and is more intuitively
appealing.
 The transfer function of the same is
shown here.
 The soft thresholding operator is
defined as
D(U, λ) = sgn(U)max(0, |U| - λ)
 Soft thresholding shrinks
coefficients above the threshold in
absolute value.
 The transfer function of the same
is shown here.

61.  The threshold
(N being the signal length, σ being the noise
variance) is well known in wavelet literature as
the Universal threshold.
σ
λ N
UNIV ln
2
=
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62.  Apply Donoho’s universal threshold,
 M is the number of pixels.
 The threshold is usually high, overly smoothing.
M
log
2
σ
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63.  Subband adaptive, a different threshold is
calculated for each detail subband.
 Choose the threshold that will minimize the
unbiased estimate of the risk:
 This optimization is straightforward, order the
wavelet coefficients in terms of magnitude and
choose the threshold as the wavelet coefficient
that minimizes the risk.
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64.  Adaptive data-driven thresholding method
 Assume that the wavelet coefficients in each
subband is distributed as a Generalized Gaussian
Distribution (GGD)
 Find the threshold that minimized the Bayesian
risk.
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65. WAVELET TOOLBOX
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66. Function Name Purpose
 dwt2 - Single-level
decomposition
[cA, cH,cV,cD] = dwt2(X, 'wname')
 Wavedec2 - Multilevel 2-D wavelet
decomposition
[C,S] = wavedec2(X, N, 'wname')
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67.  [cA1,cH1,cV1,cD1] = dwt2(X,'bior3.7');
 This generates the coefficient matrices of the
level-one approximation (cA1)
 and horizontal, vertical and diagonal details
(cH1,cV1,cD1, respectively).
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68.  Type:
 [C,S] = wavedec2(X,2,'bior3.7');
 where X is the original image matrix, and 2 is the level of
decomposition.
 The coefficients of all the components of a second-level
decomposition (that is, the second-level approximation
and the first two levels of detail) are returned
concatenated into one vector, C. Argument S is a
bookkeeping matrix that keeps track of the sizes of each
component.
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69.  Function Name Purpose
 detcoef2 - Extraction of detail
coefficients
D = detcoef2(C, S, 'wname', N)
[H,V,D] = detcoef2('all', C,S,N)
 appcoef2 - Extraction of
approximation coefficients
A = appcoef2(C,S,’wname’,N)
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70.  To extract the level 2 approximation coefficients
from C:
 cA2 = appcoef2(C,S,'bior3.7',2);
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71.  cH2 = detcoef2('h',C,S,2);
 cV2 = detcoef2('v',C,S,2);
 cD2 = detcoef2('d',C,S,2);
 cH1 = detcoef2('h',C,S,1);
 cV1 = detcoef2('v',C,S,1);
 cD1 = detcoef2('d',C,S,1);
[or]
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72.  [cH2,cV2,cD2] = detcoef2('all',C,S,2);
 [cH1,cV1,cD1] = detcoef2('all',C,S,1);
 where the first argument ('h', 'v', or 'd') determines
the type of detail
 (horizontal, vertical, diagonal) extracted, and the
last argument determines the level.
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73. Function Name Purpose
 ddencmp - Provide default values for
denoising and compression
[THR,SORH,KEEPAPP,CRIT] = ddencmp(IN1,IN2,X)
 wdencmp - Wavelet de-noising and
compression
[XC,CXC,LXC,PERF0,PERFL2] =
wdencmp('gbl',X,'wname',N,THR,SORH,KEEPAPP)
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74.  Function Name Purpose
wthcoef2 Wavelet coefficient
thresholding 2-D
NC = wthcoef2('type',C, S, N, T, SORH)
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75. Function Name
Purpose
 idwt2 - Single-level reconstruction
X = idwt2(cA, cH, cV,cD, 'wname')
 waverec2 - Full reconstruction
X = waverec2(C,S, 'wname')
 wrcoef2 - Selective reconstruction
X = wrcoef2('type',C,S,'wname',N)
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76.  To find the inverse transform:
Xsyn = idwt2(cA1,cH1,cV1,cD1,'bior3.7');
 This reconstructs or synthesizes the original
image from the coefficients of the level 1
approximation and details.
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77.  To reconstruct the level 2 approximation from C:
 A2 = wrcoef2('a',C,S,'bior3.7',2);
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78.  To reconstruct the level 1 and 2 details
from C, type:
 H1 = wrcoef2('h',C,S,'bior3.7',1);
 V1 = wrcoef2('v',C,S,'bior3.7',1);
 D1 = wrcoef2('d',C,S,'bior3.7',1);
 H2 = wrcoef2('h',C,S,'bior3.7',2);
 V2 = wrcoef2('v',C,S,'bior3.7',2);
 D2 = wrcoef2('d',C,S,'bior3.7',2);
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79.  To reconstruct the original image from the wavelet
decomposition structure:
 X0 = waverec2(C,S,'bior3.7');
 This reconstructs or synthesizes the original
image from the coefficients C of the multilevel
decomposition.
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80. GUI FOR WAVELET ANALYSIS
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81.  Starting the 2-D Wavelet Analysis Tool.
 From the MATLAB prompt, type:
wavemenu
 The Wavelet Tool Main Menu appears
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83.  Click the Wavelet 2-D menu item.
 From the File menu, choose the Load Image option.
 Load the required Image
 ANALYZING AN IMAGE
 Using the Wavelet and Level menus located to
the upper right, determine the wavelet family, the
wavelet type, and the number of levels to be used for the
analysis.
 Click the Analyze button.
 Click on any decomposition component in the lower right
window.
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84.  Click the Visualize button.
 Using Tree Mode Features.
Choose Tree from the View Mode menu.
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85.  Saving Information to Disk
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87.  From the Select thresholding method menu,
choose any item.
 Set the thresholding mode.
 Use the Sparsity slider to adjust the threshold
value
 Click the De-noise button.
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88. ◦Determination of a global optimal
threshold
◦ Spatially adjusting threshold based on
local statistics
Challenges with wavelet
thresholding
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89.  It’s possible to remove the noise with little
loss of details.
 The idea of wavelet denoising based on
the assumption that the amplitude, rather
than the location, of the spectra of the
signal to be as different as possible for
that of noise.
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90. THANK YOU
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