Chapter 3 Image Enhancement in the Spatial Domain

Chapter 3 Image Enhancement in the Spatial Domain Yinghua He School o Computer Science and Technology Tianjin University

Image enhancement approaches Spatial domain image plane itsel Spatial domain methods Based on direct manipulation o pixels in an image. Frequency domain methods Based on modiying the Fourier transorm o an image

Outline Background Some Basic Gray Level Transormations Histogram Processing Enhancement Using Arithmetic/Logic Operations Basics o Spatial Filtering Smoothing Spatial Filters Sharpening Spatial Filters Combining Spatial Enhancement Methods

Spatial domain:the aggregate o pixels composing an image. Spatial domain methods: procedures that operate directly on these pixels. Spatial domain processes: denoted by the expression: [ ( x, )] g ( x, y) = T y (x,y):the input image g(x,y):the processed image T: an operator on, deined over some neighborhood o (x,y).

s = T (r) Gray-level transormation unction o the orm r: the gray level o (x,y) s: the gray level o g(x,y) Larger neighborhoods masks:is a small 2-D array. the mask coeicients determine the nature o the process, such as image sharpening. Masking processing Enhancement techniques based on this type o approach oten are reerred to as masking processing.

Three basic types o unctions used requently or image enhancement: Linear (negative and identity transormations) Logarithmic (log and inverse-log transormations) Power-law (nth power and nth root transormations)

Image negatives The negative o an image with gray levels in the range[0,l-1] is obtained by using the negative transormation in Fig. 3.3, which is given by the expression s=l-1-r

Log Transormations The general orm o the log transormation shown in Fig. 3.3 is s = c log( 1+ r) where c is a constant, and it is assumed that r>=0. This transormation maps a narrow range o low gray-level values in the input image into a wider range o output levels.

Power-Law Transormations Power-law transormations have the basic orm γ γ s = cr Where c and are positive constants.

2.5 s = s = r 1/ 2. 5 r

Piecewise-Linear Transormation Functions

Arithmetic/logic operations involving images are perormed on a pixel-by-pixel basis between two or more images. O the our arithmetic operations, subtraction and addition are the most useul or image enhancement. Masking sometimes is reerred to as region o interest(roi) processing.

Image subtraction Image Averaging

Image subtraction g( x, y) = ( x, y) h( x, y)

Image subtraction Image Averaging

Consider a noisy image g(x,y) ormed by the addition o noise to an original image (x,y); that is I an image is ormed by averaging K dierent noisy images, ), ( y x η ), ( ), ( ), ( y x y x y x g η + = ), ( y x g = = K i i y x g K y x g 1 ), ( 1 ), (

Then it ollows that And { g( x, y) } ( x, y) E = σ = 2 g ( x, y) 1 K σ 2 η ( x, y)

Some neighborhood operations work with the values o the image pixels in the neighborhood and the corresponding values o a subimage that gas the same dimensions as the neighborhood. The values in a ilter subimage are reerred to as coeicients, rather than pixels. For linear spatial iltering, the response is given by a sum o products o the ilter coeicients and the corresponding image pixels in the area spanned by the ilter mask.

The result, R, o linear iltering with the ilter mask at a point(x,y) in the image is R = w( 1, 1) ( x 1, y 1) + w( 1,0) ( x 1, y) +... + w(0,0) ( x, y) +... + w(1,0) ( x + 1, y) + w(1,1) ( x + 1, y + 1) In general, linear iltering o an image o size M*N with a ilter mask o size m*n is given by the expression: a b g ( x, y) = w( s, t) ( x + s, y + t) s= at= a

When interest lies on the response, R, o an m*n mask at any point(x,y), it is common pratice to simpliy the notation by using the ollowing expression: R = w1 z1 + w2 z 2 +... + w mn z mn = mn i= 1 w i z i

For the 3*3 general mask shown below, the response at any point(x,y) in the image is given by

I the center o the mask moves any closer to the border, one or more rows or columns o the mask will be located outside the image plane. There are several ways to handle this situations. To limit the excursions o the center o the mask to be at a distance no less than (n-1)/2 pixels rom the border. To ilter all pixels only with the section o the mask that is ully contained in the image. Padding the image by adding rows and columns o 0 s, or padding by replicating rows and columns.

Smoothing ilters are used or blurring and or noise reduction. Blurring is used in preprocessing steps, such as removal o small details rom an image prior to object extraction, and bridging o small gaps in lines or curves. Noise reduction can be accomplished by blurring with a linear ilter and also by non-linear iltering.

Smoothing Linear Filter Order-Statistics Filter

The output o a smoothing, linear spatial ilter is simply the average o the pixels contained in the neighborhood o the ilter mask. This ilters sometimes are called averaging ilters. They also reerred to a lowpass ilter.

The general implementation or iltering an M*N image with a weighted averaging ilter o size m*n is given by the expression = = 9 1 9 1 i z i R = = = = + + = a a s b b t a a s b b t t s w t y s x t s w y x g ), ( ), ( ), ( ), (

Smoothing Linear Filter Order-Statistics Filter

Order-statistics ilters are nonlinear spatial ilters whose response is based on ordering (rankng) the pixels contained in the image area encompassed by the ilter, and then replacing the value o the center pixel with the value determined by the ranking result. The best-known example in this category is the ilter, which, as its name implies, replaces the value o a pixel by the median o the gray levels in the neighborhood o that pixel.

The principal objective o sharpening is to highlight ine detail in an image or to enhance detail that has been blurred, either in error or as a natural eect o a particular method o image acquisition. Averaging is analogous to integration; Sharpening could be accomplished by spatial dierentiation.

Foundation Use o Second Derivatives or Enhancement- The Laplacian Use o First Derivatives or Enhancement- The Gradient

We require that any deinition we use or a irst derivative. Must be zero in lat segments(areas o constant gray-level values); Must be nonzero at the onset o a gray-level step or ramp; Must be nonzero along ramps.

The deinition o a second derivative Must be zero in lat areas; Must be nonzero at the onset and end o a graylevel step or ramp; Must be zero along ramps o constant slope.

A basic deinition o the irst-order derivative o a one-dimensional unction (x) is the dierence. x = ( x + 1) ( x) We deine a second-order derivative as the dierence. 2 x 2 = ( x + 1) + ( x 1) 2 ( x)

Comparing the response between irst- and secondorder derivatives, we arrive at the ollowing conclusions. First-order derivatives generally produce thicker edges in an image. Second-order derivatives have a stronger response to ine detail, such as thin lines and isolated points. First-order derivatives generally have a stronger response to a gray level step. Second-order derivatives produce a double response at step changes in gray level.

Second-order derivatives that, or similar changes in gray-level values in an image, their response is stranger to a line than to a step, and to a point than to a line.

Foundation Use o Second Derivatives or Enhancement- The Laplacian Use o First Derivatives or Enhancement- The Gradient

We are interested in isotropic( 同向性 ) ilters, whose response is independent o the direction o the discontinuities in the image to which the ilter is applied. Isotropic ilters are rotation invariant.

The simplest isotropic derivative operator is the Laplacian, which, or a unction (image)(x,y) o two variables is deined as 2 2 2 = + 2 2 x y Because derivatives o any order are linear operations, the Laplacian is a linear operator.

We use the ollowing notation or the partial second-order derivative in the x-direction: And, similarly in the y-direction, as ), ( 2 ) 1, ( ) 1, ( 2 2 y x y x y x x + + = ), ( 2 1), ( 1), ( 2 2 y x y x y x y + + =

The digital implementation o the twodimensional Laplacian is obtained by summing these two components. [ ] ), ( 4 1), ( 1), ( ) 1, ( ) 1, ( 2 y x y x y x y x y x + + + + + =

Because the Laplacian is a derivative operator, its use highlights gray-level discontinuities in an image and deemphasizes regions with slowly varying gray levels. The basic way in which we use the Laplacian or image enhancement is as ollows: g( x, y) = ( x, ( x, y) y) + 2 2 ( x, ( x, y) y) i the center coeicient o Laplacian mask is negative i the center coeicient o Laplacian mask is positive the the

Simpliications [ ] [ ] 1), ( 1), ( ) 1, ( ) 1, ( ), ( 5 ), ( 4 1), ( 1), ( ) 1, ( ) 1, ( ), ( ), ( + + + + + = + + + + + + = y x y x y x y x y x y x y x y x y x y x y x y x g

Unsharp masking and highyboost iltering A process used or many years in the publishing industry to sharpen images consists o subtracting a blurred version o an image rom the image itsel. This process, called unsharp masking, is expressed as s ( x, y) = ( x, y) ( x, y) Where s ( x, y) denotes the sharpened image obtained by unsharp masking and ( x, y) is a blurred version o ( x, y).

A slight urther generalization o unsharp masking is called high-boost iltering. A highboost iltered image, hb, is deined at any point (x,y) as hb ( x, y) = A ( x, y) ( x, y) where A>=1 The equation may be written as hb ( x, y) = ( A 1) ( x, y) ( x, y) s

I we select to use the Laplacian, the above equation becomes 2 i the center coeicient o the A ( x, y) ( x, y) Laplacian mask is negative hb = 2 i the center coeicient o the A ( x, y) + ( x, y) Laplacian mask is positive When A=1, high-boost iltering becomes standard Laplacian sharpening.

For unction (x,y), the gradient o at coordinates (x,y) is deined as the twodimensional column vector The magnitude o this vector is given by = = y x G G y x [ ] 2 1 2 2 2 1 2 2 ) ( + = + = = y x G G mag y x

In order to reduce the computational burden, it is common practice to approximate the magnitude o the gradient by using absolute values instead o squares and square roots: G x + G y

The simplest approximations to a irst-order derivative that satisy the conditions stated are = z 8 z ) and = z 6 z ). G x ( 5 G y ( 5 The other deinitions proposed by Roberts in the early development o digital image processing use cross dierences: and = z 8 z G x = z 9 z ) ) ( 5 G y ( 6

We compute the gradient as or An approximation using absolute values, still at point, but using a 3*3 mask, is [ ] 2 1 2 6 8 2 5 9 ) ( ) ( z z z z + = 6 8 5 9 z z z z + z 5 ) 2 ( ) 2 ( ) 2 ( ) 2 ( 7 4 1 9 6 3 3 2 1 9 8 7 z z z z z z z z z z z z + + + + + + + + +

Sobel operators Roberts crossgradient operators