Image Enhancement

chương 1: xử lý ảnh

DIGITAL IMAGE PROCESSING

3.1 Introduction

Objective of Image Enhancement : to process an image so that the processed image

is more suitable than the original image for a specific application

Relation of Enhancement and Restoration : in image restoration, an original image has been degraded and the objective is to make the processed image resemble the original image as much as possible

Classification of Enhancement Techniques

Point Operations : on each pixel, the input gray level is mapped into a new one

Sapatial Operations : spatial filtering

Transform Operations : manipulation in transform domain

¢ Clipping : a special case of contrast stretching where a =y =0 (Fig 3.2.1)

useful for noise reduction of the input signal known to lie in fa, 5]

should be performed on images that will be represented with a finite number of bits, for example, with unsigned character

Department of Biomedical Engineering

Other Simple Operations

« Image negative: v=L—u, u,ve [0,L]

Fig 3.2.3 (a)(c) Original images, (b)(d) negative images

« Range Compression: The dynamic range of a unitarily transformed image is so

large that only a few pixels are visible The dynamic range can be compressed via

the logarithmic transformation

v=clog,(I+|w|) where c isascaling constant

« Image Subtraction: In many imaging applications it is desired to compare two

images A simple but powerful method is to align the two images and subtract them

Digital Image Processing Image Enhancement

Histogram Processing

« Histogram of a Digital Image : represents the relative frequency of occurrence of

the various gray levels in the image The histogram gives an estimate of the

probability of occurrence of each gray level

« Histogram Equalization : The goal is to obtain a uniform histogram for the output

image Consider an image pixel value u20 to be a random variable with a

continuous pdf p(w) andcdf F (6)=P[u <6] Then the random variable

will be uniformly distributed over (0,1)

(Proof) From Eq (3.2.1), the derivative of v with respect to wu is

du Substituting Eq (3.2.2) into the relation of p,(v) and p,(u):

(Implementation of histogram equalization on digital image)

Suppose the input uw hasa histogram A(x,), 7=0,1, -,2—1 Then we obtain

Fig 3.2.5 Histogram equalization transformation

Image Enhancement

Digital Image Processing

(Example) Histogram equalization for the given histogram A(u) of a 3-bit image

Fig 3.2.6 Histogram equalization (a) original image, (b) original histogram (x-axis:

[0,255], y-axis: [0, 255]), (c) equalized image, (d) equalized histogram

« Histogram Specification : Suppose the random variable ¿>0 with pdf p,(w)

is to be transformed to v2=0 such that it has a specified pdf p,(v) For this to be

true, we define a uniform random variable

Fig 3.2.7 Histogram specification

(Example) Histogram specification for the given histograms (pdf) p,(w) and p.(v) of a 2-bit ima

3.3 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 prior to (large)

object extraction, and bridging of small gaps in lines or curves

Spatial Averaging

¢ Each pixel is replaced by a weighted average of its neighborhood pixels, that 1s,

vữứn,n)= 3S a(k,l)y(m—k,n—]) (3.3.1)

(kJ)

where y(m,n) and v(m,n) are the input and output images, respectively, W is

a suitably chosen window, and a(k,/) are the filter weights {a(k,/)} is an

impulse response called spatial mask A common class of spatial averaging filters

has all equal weights, giving

l

we (LJ

where a(k,/)=1/N, and N,, is the number of pixels in the window W Fig

3.3.1 shows some spatial averaging masks

Digital Image Processing Image Enhancement

Fig 3.3.1 Spatial averaging masks

Spatial averaging is used for noise smoothing, low-pass filtering, and sub-sampling

of images Suppose the observed image ts given as

where 1(m,n) is white noise with zero mean and variance O° Then the spatial

average of Eq (3.3.3) yields

(tlie

4

where T[Ún,m) is the spatial average of †J(m.n) It can be shown that TJm,n) has

Zzcro mean and varlance ỞØ ` =Ø”/N,, that ¡s, the noise power is reduced by a

factor equal to the number of pixels in the window W

Fig 3.3.3 (a) Original image, (b) — (d) results of spatial filtering

with a mask of Fig 3.3.1 (a) —(c)

Digital Image Processing Image Enhancement

Median Filtering

The input pixel is replaced by the median of the pixels contained in a window W

around the center pixel, that 1s,

v(m, n) = median { y(m—k,n—-1), (k, lye W} (3.3.5)

If N, is even, then the median is taken as the average of the two values in the

middle

Example 3

Let { y(m)} ={2,3,8,4,2} and W =[-1,0,1] The median filter output is given by

v(0)=2 (boundary value), v(1) = median} 2,3,8} = 3 v(2) = median} 3,8,4} = 4, v(3) = median{S.4.2} = 4 v(4) =2 (boundary value)

Hence {v(m)}= {2,3,4,4,2} If W contains an even number of pixels — for

example, W =[-1,0,1,2] — then v(0)=2, v(1l)=3, v(2) =median{2,3.8,4} =3.5,

v(3) = median{3,8,4,2} = 3.5, and v(4)=2 gives {v(m)} = {2,3,3.5,3.5,2}

Properties of median filter

1 Itis a nonlinear filter, that is,

median x(m) + y(m)} # median x(m)} + median y(m)}

2 It reduces impulsive noise well and also preserves edges well

Fig 3.3.5 (a) Original image, (b) image with binary noise (-128 and 128 for 10 %),

(c) averaging with 3x3 mask, and (d) 3x3 median filtered image

Digital Image Processing Image Enhancement

Sharpening (Crispening)

e« Psychophysical experiments indicate that a photograph or visual signal with

accentuated or crispened edges is often more subjectively pleasing than exact

photometric reproduction

«Ắ 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 from the image This is

equivalent to adding the gradient, or a high-pass signal, to the image (See Fig

3.3.7) The unsharp masking operation can be represented by

v(m.n) =(A +1)ju(m,n)—Ah,,u(m.n)}, A>O

=u(m,n)+Alu(m,n)—h,,u(m,n)] (3.3.6)

=u(m,n)+Ah,,u(m,n)

where h,, and h,, mean low-pass and high-pass filters, respectively The

constant A is typically chosen as 0.25 ~ 0.33 Eq (3.3.6) also called high-boost or

high-frequency-emphasis filtering

Digital Image Processing Image Enhancement

« Statistical Differencing : The statistical differencing, suggested by Wallis,

forces the enhanced image to a form with desired mean and standard deviation The

operation is defined by

oO

v(m,n) G.x) £ữÐ [w(m.m) — H(m.n)]+[ mu +(1— B)H(m.n)]} — ( )

where O(m,n) and H(m,m) represent local mean and standard deviation, oO,

and m, denote desired mean and standard deviation, @ is a gain factor that

prevents overly large output values when O(m,n) is small, and B is a factor

controlling the ratio of the edge to background intensities The constant O,, m d # 3

œ,and are typically chosen as 8.5, 128, 1/6, and 0,1

Fig 3.3.11 (a)(c) Original image, (b)(d) images after statistical differencing operation

as

where i(m,n) represents the illumination and r(m,n) represents the reflectance

Assumption : i(m,n) 1s primary contributor to the dynamic range, varying slowly

r(m,n) is primary contributor to local contrast, varying rapidly

To separate i(m,n) from r(m,n), a logarithmic operation is applied to Eq (3.3.8)

logu(m,n) = logi(m,n) + logr(m,n) (3.3.9)

Low-pass filtering logu(m,n) — logi(m,n) High-pass filtering logu(m,n) — logr(m,n)

Digital Image Processing Image Enhancement

logi(m,n) is attenuated to reduce the dynamic range while logr(m,n) is emphasized to increase the local contrast The processed logi(m,n) and

logr(m,n) are then combined and the result is exponentiated to get back to the

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 This operation is described in Fig

Fig 3.3.14 Description of digital zooming by continuous interpolator

The continuous signal interpolated 1s given as

where A (x) denotes a continuous interpolation function

Digital Image Processing Image Enhancement

1 Sync-Function Interpolator (not spatially limited):

sin 70x 7x

The above interpolation functions are shown in Fig 3.3.15

e L:1 Interpolator

Based on the theory of digital signal processing, the L:1 interpolator which

increases sampling rate by L consists of upsampler, low-pass filter, and amplifier

with gain of L It is shown in Fig 3.3.16

Fig 3.3.16 Block diagram of L:1 interpolator

Digital Image Processing Image Enhancement

The upsampler produces a sequence

(m) u(m/L), if m is integer multiple of LZ

m)=

=u([m/ L]) Then the output sequence v(m) can be written as

v(m) = L¥ h(k) y(m—k) = LY h(k)u({(m— k)/ L)) (3.3.13)

Where the interpolation filter A(k) in general is a symmetric LPF and }{A(k)=1

For example, any QMF filter can be selected as a good interpolation filter

Relation between continuous interpolator and L:1 interpolator

Setting x =m/ L, the Eq (3.3.10) becomes

The decimator sequence v(m) can be written as

Fig 3.3.18 Results by 2:1 Interpolators (a) square, (b) triangular,

(c) bell, and (d) cubic B-spline

Digital Image Processing Image Enhancement

V (k,l)

« Image enhancement by transform filtering

——+» transformation }+——% operations +=} _ transformation }———_»

Fig 3.4.1 Image enhancement by transform filtering