[Ch6] Spatial Filtering

6.1 Introduction

1) Spatial Filtering

Spatial Filtering  : applying a function to a neighborhood of each pixel

① Move a 'mask' : a rectangle (usually with sides of odd length) or other shape over given img

② Create a new img whose pixels have grey values calculated from grey values under the mask

 

2) Linear Filter

Filter : Combination of Mask and Function

Linear Filter : Combination of Mask and Linear function

Multiplying all elements in Mask

by corresponding elements in Neighborhood Position  

→ Adding up all products

 

 

3) Process for performing spatial filtering (3 steps)

(1) Position the Mask over the current pixel 

(2) Form all products of filter elements with corresponding elements of neighborhood

(3) Add up all the products

(1)~(3) must be repeated for every pixel in img

 

[Ex] Averging filtering

(1) 3x3 Average Mask

- e : grey value of the current pixel in the original img

- Average : grey value of the corresponding pixel in the new img

 

(2) 5x5 img

>> x=uint8(10*magic(5))

x =

  5×5 uint8 행렬

   170   240    10    80   150
   230    50    70   140   160
    40    60   130   200   220
   100   120   190   210    30
   110   180   250    20    90

 

(3) Result of filtering 5x5 img with 3x3 Averaging filter

 

 

 

6.2 Notation

1) Matrix of Linear Filter

[Ex] Another Filter

 

2) Edges of the image

(1) Obvious problem in applying a filter at the edge of img

: Lack of grey value to use in the filter function

 

(2) Approaches to solve this problem

① Ignoring the edges

: Applying the mask to those pixels in img for with the mask will lie fully within img

∴ Output img size < Original img size

+) If mask is very large, we may lose a significant amount of information

 

② Zero Padding

: Assuming that all necessary values outside th img are Zero

∴ Output img size = Original img size

+) But, Unwanted artifacts (ex. edges) around the img can happen

 

 

6.3 Filtering in MATLAB

1) How to make Filter

(1) Creating filters by hand

>> a=ones(3,3)/9

a =

    0.1111    0.1111    0.1111
    0.1111    0.1111    0.1111
    0.1111    0.1111    0.1111

 

(2) Using MATLAB function : fspecial('<option>', <filter size>)

🍏 Many options to make for easy creation of many different filters

  Ex. 'average' option for blurring img

>> f1=fspecial('average')       # 3x3 size filter (default)
>> f2=fspecial('average',[5,7]) # 5x7 size filter
>> f3=fspecial('average',[11])  # 11x11 size filter

>> h = fspecial(type)
>> h = fspecial('average',hsize)
>> h = fspecial('disk',radius)
>> h = fspecial('gaussian',hsize,sigma)
>> h = fspecial('laplacian',alpha)
>> h = fspecial('log',hsize,sigma)
>> h = fspecial('motion',len,theta)
>> h = fspecial('prewitt')
>> h = fspecial('sobel')

'average'	평균 필터
'disk'	원형 평균 필터(필박스)
'gaussian'	가우스 저역통과 필터 (권장 : imgaussfilt 또는 imgaussfilt3)
'laplacian'	2차원 라플라시안 연산자 근사 필터
'log'	LoG(가우스-라플라시안) 필터
'motion'	카메라의 선형 움직임 근사 필터
'prewitt'	프루이트(Prewitt) 가로 방향 경계 강조 필터
'sobel'	소벨(Sobel) 가로 방향 경계 강조 필터

 

미리 정의된 2차원 필터 생성 - MATLAB fspecial - MathWorks 한국

 

 

🍏 3 different ways to display a matrix of data type double

① Transform it to a matrix of type uint8, for use with imshow

② Divide its values by 255 to obtain a matrix with values in 0.1~1.0, for use with imshow

③ Use mat2gray to scale the result for display

>> c=imread('cameraman.tif');
>> f1=fspecial('average');
>> cf1=filter2(f1,c);

>> figure,imshow(c),figure,imshow(cf1/255)

 

- Further blurred by using an averaging filter of larger size

 

 

2) Filtering MATLAB function

(1) filter2(filter, image, shape) for Linear filtering

- shape : optional parameter to describe method for dealing with the edges

- Result : A matrix of data type double

 

(2) Examples

▪ filter2(filter, image, 'same') ; default

- How : Using zero padding

- Result : A matrix of equal size to original

>> filter2(a,x,'same')

ans =

   76.6667   85.5556   65.5556   67.7778   58.8889
   87.7778  111.1111  108.8889  128.8889  105.5556
   66.6667  110.0000  130.0000  150.0000  106.6667
   67.7778  131.1111  151.1111  148.8889   85.5556
   56.6667  105.5556  107.7778   87.7778   38.8889

 

▪ filter2(filter, image, 'valid')

- How : Applying the mask only to inside pixels

- Result : A matrix of smaller size than original

- Large filter 사용 시 : Zero padding used at the edges 에 의해 dark border appearing around tmg = unwanted artifact 생길 수 있음 (ex. changing the average brightness of img) → 'valid' shape option 사용하면 방지 가능 ! 

>> filter2(a,x,'valid')

ans =

  111.1111  108.8889  128.8889
  110.0000  130.0000  150.0000
  131.1111  151.1111  148.8889

 

▪ filter2(filter, image, 'full')

- How : Using zero padding & Applying the filter at all places on and around img where the mask intersects img

- Result : A matrix of larger size than original

>> filter2(a,x,'full')

ans =

   18.8889   45.5556   46.6667   36.6667   26.6667   25.5556   16.6667
   44.4444   76.6667   85.5556   65.5556   67.7778   58.8889   34.4444
   48.8889   87.7778  111.1111  108.8889  128.8889  105.5556   58.8889
   41.1111   66.6667  110.0000  130.0000  150.0000  106.6667   45.5556
   27.7778   67.7778  131.1111  151.1111  148.8889   85.5556   37.7778
   23.3333   56.6667  105.5556  107.7778   87.7778   38.8889   13.3333
   12.2222   32.2222   60.0000   50.0000   40.0000   12.2222   10.0000

 

 

 

6.4 Frequencies ; low and high pass filters

1) Frequencies

: the amount by which grey values change with distance

- one important aspect of an img which enables us to choose the most appropriate filter

 

(1) High freq components

: characterized by large changes in grey values over small distances

Ex.  Edges, Noise

 

(2) Low freq components

: characterized by little changes in grey values

Ex. Background, skin textures

 

2) Filters

(1) High Pass Filter

: passing over the high freq components & reducing or eliminating low freq components

- The sum of the coefficients(all elements in filter matrix) = 0

- Applying HPF to Low freq part of img(where grey values are similar) : resulting grey values → 0

- Used for Edge Detection, Edge Enhancement

Ex. Laplacian filter, Laplacian of Gaussian ("log") filter

>> f=fspecial('laplacian')

f =
    0.1667   0.6667   0.1667
    0.6667  -3.3333   0.6667
    0.1667   0.6667   0.1667
    
>> cf=filter2(f,c);
>> imshow(cf/100)


>> f1=fspecial('log')

f1 =
    0.0448   0.0468   0.0564   0.0468   0.0448
    0.0468   0.3167   0.7146   0.3167   0.0468
    0.0564   0.7146  -4.9048   0.7146   0.0564
    0.0468   0.3167   0.7146   0.3167   0.0468
    0.0448   0.0468   0.0564   0.0468   0.0448

>> cf1=filter2(f1,c);
>> figure,imshow(cf1/100)

 

(2) Low Pass Filter

: passing over the low freq components & reducing or eliminating high freq components

Ex. 3x3 averaging filter (blurring edges)

 

 

3) How to deal with values outside 0-255

For img display, grey values of pixels to lie bw 0-255

But, result of linear filtering values lie outside 0-255

(1) Make Negative values Positive

- certainly dealing with negative values, but not with values greater than 255

- So, (1) can only be used in specific circumstances

  (Ex. only a few negative values & these values are close to zero)

 

(2) Clip values

- applying thresholding type operation to resulting grey value x to obtain displayable value y :

- producing img with all pixel values in required range

- But, not suitable if there are many grey values outside 0-255 range

  (In particular, equally spread over a large range)

  → Destroy the results of filtering 

 

(3) Scaling transformatioins

- x축 : g_L : the lowest grey value produced by filter / g_H : the highest grey value

- y축 : transforming all value in g_L - g_H to 0 - 255 by linear transformation :

 

 

→ 0 ≤ y ≤ 255

 

 

 

 

>> f2=[1 -2 1;-2 4 -2;1 -2 1];    # HPF
>> cf2=filter2(f2,c);             # cf2 : -541~593
>> figure,imshow(mat2gray(cf2));  # mat2gray -> double(0~1)

>> maxcf2=max(cf2(:));
>> mincf2=min(cf2(:));
>> cf2g=(cf2-mincf2)/(maxcf2-mncf2); # linear transformation -> uint8(0~255)
>> figure,imshow(cf2/60)    # better result by dividing result by a constant

 

6.5 Gaussian filters

1) Gaussian filters

Gaussian filters : Low Pass Filters, all based on the Gaussian probability distribution function

- blurring effect similar to neighborhood averaging 

- σ : standard deviation (large σ produces to a flatter curve, small σ leads to a pointier curve)

 

(1) One-dimensional Gaussian function

 

(2) Two-dimensional Gaussian function

 

2) MATLAB function 

▪ fspecial('gaussian') : producing Discrete version of Gaussian function

- second parameter (default=3) : filter size (square이면 single number 가능)

- final parameter (default=0.5) : standard deviation  σ (클수록 평평)

 

▪ surf(1:a, 1:a, g) : drawing pictures of Discrete Gaussian function

>> a=50;s=3;
>> g=fspecial('gaussian',[a a],s);
>> surf(1:a,1:a,g)

>> s=9;
>> g2=fspecial('gaussian',[a a],s);
>> figure,surf(1:a,1:a,g2)

>> g1=fspecial('gaussian',[5,5]);
>> g2=fspecial('gaussian',[5,5],2);
>> g3=fspecial('gaussian',[11,11],1);
>> g4=fspecial('gaussian',[11,11],5);

>> imshow(filter2(g1,c)/256)
>> figure,imshow(filter2(g2,c)/256)
>> figure,imshow(filter2(g3,c)/256)
>> figure,imshow(filter2(g4,c)/256)

 

3) Effect of Large Standard deviation without bound

- Result : Averaging filter as limiting values !

- Gaussian blurring과 Averaging 결과는 비슷해보임

- But, Gaussian filter : elegant mathematical properties 가져서 suitable for blurring

>> fspecial('gaussian.,3,100)

ans =
    0.1111 0.1111 0.1111
    0.1111 0.1111 0.1111
    0.1111 0.1111 0.1111

 

6.6 Non-linear filters

non-linear filters can be used, if less efficiently

 

1) nlfilter(<img>, <filter size>, <function>)

: Applying a filter to an img according to a pre-defined function

  (If funtion is not already defined, we have to create an m-file which defines it)

 

- 3 Arguments : img matrix, filter size, matrix function which returns a scalar value

- Result : img has lost some sharpness & been brightened by maximum filter and darkend by minimum filter

- Very slow → little call for non-linear filters

>>> cmax=nlfilter(c,[3,3],'max(x(:))');
>>> cmin=nlfilter(c,[3,3],'min(x(:))');

 

2) ordfilt2

: more efficient MATLAB function