ASAM Image Processing 20082009

1
ASAM - Image Processing2008/2009

• Lecture 21
• Revision lecture III

Ioannis Ivrissimtzis 13-May-2009
2
Revision questions lecture 12

• i. Describe the method of intensity slicing for
  pseudo-colouring.
• ii. Describe gray to colour transformations and
  image filtering for pseudo-colouring.
• iii. Describe colour transformations and give one
  example for each of the RGB, CMY and HSI colour
  spaces.

3
Intensity slicing

• A simple way to produce a pseudo-colouring is the
  intensity slicing.
• We consider the intensity levels
  
• where corresponds to black and to
  white.
• We use the M colours

4
Intensity slicing

• A pixel with intensity s is assigned the colour
  if
• The intensity is assigned the colour
  .

5
Example of intensity slicing

• Monochrome image of the Picker Thyroid Phantom

Results of intensity slicing into eight colour
regions
Images from Gonzalez and Woods
6
Gray to colour transformations

• The gray level to colour transformations are
  defined by three functions from the set of gray
  levels to the sets of intensities of the red,
  green and blue bands respectively.
• This transformation can be seen as consisting of
  three independent intensity transformation as
  described in Lecture 2 of this course.

7
Gray to colour transformations

• The three intensity transformations
• describe a gray level to colour transformation
• A pixel with intensity s gets the RGB colour

8
Image filtering for pseudo-colouring

• In a different approach to pseudo-colouring we
  compute three independent transformations of the
  image rather than the gray scale.
• For a grayscale image we compute
  its transforms
• The pseudo-coloured image is written in the RGB
  space as

9
Example
Notice that in the pseudo-coloured image the
outer ring of Saturn is much more visible.
The red band with high frequency information
Original grayscale image
Pseudo-coloured
10
Colour transformations

• Similarly to the intensity transformations for
  grayscale images and gray to colour
  transformations for pseudo-colouring, we have
  colour transformations.
• In the RGB space for example, under a colour
  transformation, each RGB colour
• is mapped to a colour

11
RGB example

• The colour transformation
• inverses the colours in a way reminiscent of the
  negatives of the colour films.

12
RGB example
Original image
RGB negative
13
CMY example

• Visual inspection of the image shows an excess of
  magenta.
• To balance the colour we convert it to the CMY
  space and transform the magenta component.

14
CMY example
1
1
0
Transformation function of magenta
Original image heavy on magenta
Corrected image
15
HSI example

• We want to brighten the image using histogram
  equalisation.
• Histogram equalisation on each RGB component we
  will change the hues and the processed image will
  look unnatural.

16
HSI example

• Instead we
• Convert the image to the HSI space.
• We transform the intensity component applying
  histogram equalisation on it.
• In addition, we transform the saturation
  component to get less saturated colours.

17
HSI example
1
1
0
Transformation function of saturation
Original image
Processed image
The transformation function of the intensity
component was computed by applying histogram
equalisation on it.
18
Revision questions lecture 13

• i. Describe the morphological operations
  dilation, erosion, opening and closing for binary
  images.
• ii. Describe how binary morphology can be used
  for boundary extraction.

19
Revision questions lecture 13

• iii. Find the number of ones at the binary image
  I after dilation, erosion, opening and closing
  with the structuring element S.

20
Structuring elements

• Similarly to the case of spatial linear
  filtering, a binary image can be processed by
  another binary image called structuring element.
• A typical structuring element has few 1-valued
  pixels, and one of his pixels is designated a its
  origin.
• In an analogy between spatial linear filtering
  and morphological image processing, the
  structuring element can be seen as the equivalent
  of the mask and its origin as the equivalent of
  the centre of a mask.

21
Dilation

• Let A be a binary image and B a structuring
  element.
• The dilation of A by B is the set consisting of
  all the locations of the origin of B where the
  translated B overlaps at least some portion of A.
  
• The mechanics of the dilation are similar to that
  of spatial linear filtering. We translate B in
  all possible positions and if there is some
  overlap with A, then the location of the origin
  belongs to the dilation.

22
Erosion

• Let A be a binary image and B a structuring
  element.
• The erosion of A by B is the set consisting of
  all the locations of the origin of B where the
  translated B fits entirely within A.
• The mechanics of the erosion are similar to that
  of dilation. We translate B in all possible
  positions and if it fits entirely within A, then
  the location of the origin belongs to the
  erosion.

23
Opening

• In practical applications, dilations and erosions
  are used most often in various combinations.
• The opening of the binary image A by the
  structuring element B is the erosion of A by B,
  followed by dilation of the result by B.

24
Closing

• The closing of the binary image A by the
  structuring element B is the dilation of A by B,
  followed by erosion of the result by B.
• We can show that the closing of A by B is the
  complement of union of all translations of B that
  do not overlap A.

25
Boundary extraction

• Let A ? B denotes the dilation of A by B and let
  A - B denotes the erosion of A by B.
• The boundary of A can be computed as
• A - ( A - B )
• where be is a 3x3 square structuring element.
• That is, we subtract from A an erosion of and
  obtain its boundary.

26
Exercise

• After dilation the image has 44 one-valued images

27
Exercise

• After erosion the image has 8 one-valued images

28
Exercise

• After opening the image has 20 one-valued images

29
Exercise

• After closing the image has 24 one-valued images

30
Revision questions lecture 14

• i. Describe the use of the following masks for
  point and line detection
• ii. Describe edge detection based on the
  Laplacian of the Gaussian.

31
Point detection

• For the detection of isolated points we can use
  the mask

It is a form of the Laplacian with the diagonal
directions included.
32
Point detection

• We say that a point has been detected at the
  location (x,y), if the absolute value of the
  response is above a threshold T.
• If the detected points are labelled 1 and all
  others are labelled 0, we can visualise the
  results of point detection by a binary image
  g(x,y)

33
Line detection

• A line is an edge segment in which the intensity
  of the background on either side is either much
  higher or much lower than the intensity of the
  line pixels.
• The Laplacian mask
• can be used for line detection.

34
Line detection

• The previous mask is isotropic. It does not have
  preference for any particular direction.
• Anisotropic masks can be used to detect lines in
  specified directions

Horizontal
45
Vertical
- 45
35
Line detection

• Let R1, R2, R3, R4 be the responses of the four
  masks centred on (x,y).
• The pixel (x,y) is said to be more likely
  associated with the direction of the mask with
  the highest response in absolute values.
• For example, if
• we say that (x,y) is more likely associated with
  the horizontal direction.

36
Laplacian of Gaussian

• Edge detectors must be able to cope with noise.
  They must also be able to act at any scale.
• The Marr-Hilderth edge detector, solves these two
  problems by first applying a Gaussian mask on the
  image, and then a Laplacian.
• The size of the Gaussian mask depends on the
  scale of the edges we want to detect.

37
Laplacian of Gaussian

• The Laplacian of the Gaussian of an image can be
  efficiently implemented using a single mask
  computed as the convolution of a Gaussian mask by
  a Laplacian mask.
• An example of a Laplacian of Gaussian mask is

38
Laplacian of Gaussian

• The final step is to compute the zero-crossings,
  that is, the pixels where the Laplacian of the
  Gaussian changes sign.
• Typically, we define a binary image with 0s
  corresponding to negative pixel values and 1 to
  positive.
• Then we compute the boundary of the binary image,
  e.g. with morphological operators.

39
Revision questions lecture 15

• i. Use the Sobel masks Sx, Sy to compute the
  gradient at the centre of the image I.
• ii. Describe the Canny edge detector.

40
Exercise
The first order difference in the
x-direction (vertical) is the response the Sobel
mask
The first order difference in the
y-direction (horizontal) is the response of the
Sobel mask
41
Exercise
The gradient at the centre of I is
I
42
Canny edge detector

• The Canny edge detection algorithm consists of
  the following steps
• Smooth the input image with a Gaussian filter
• Compute the gradient magnitude and angle
• Apply non-maxima suppression to the gradient
  magnitude
• Use double thresholding and connectivity analysis
  to detect and link edges

43
Non-maxima suppression

• In the array (image) of the gradient magnitude
  the edges of the original image are represented
  by wide ridges. See previous lecture
• The non-maxima suppression step thins these
  ridges, by keeping only the locally maximum
  values of the gradient magnitude .

44
Non-maxima suppression

• The maximum suppression first quantizes the
  gradient angle into the four directions of a
  pixels neighbourhood.

For each pixel, we find the sector of the
gradient angle, and assign to the pixel one of
the four directions, horizontal, - 45, vertical,
or - 45.
45
Non-maxima suppression

• The second step of non-maxima suppression makes
  zero the gradient magnitude of a pixel if it is
  smaller than the gradient magnitude of any of the
  two pixels in the quantized gradient direction.
• For example, if the quantized gradient direction
  is horizontal, then we compare
  with the gradient magnitude values at its left
  and its right, and make it zero if it is smaller
  than either of them.

46
Double thresholding

• We use a high and a low threshold for
  the gradient magnitude values.
• The edges are found by a tracking algorithm
  starting from pixels above the high threshold and
  stopping and pixels below the low threshold.

47
Edge tracking

• The edge tracking algorithm
• Start a new edge from a pixel above the high
  threshold that has not already been visited
• Track the edge, following the quantized gradient
  angle in both directions and mark all pixels
  above the low threshold as edges. The edge
  tracking stops at pixels below the low threshold.
  
• Until all pixels above the high threshold have
  been visited.

48
Example
Original image
Thresholded gradient of smoothed image
Result of Canny algorithm
49
Revision questions lecture 16

• i. Describe a local edge linking algorithm for
  detecting edges in the horizontal and vertical
  direction.

50
Edge linking algorithm

• Compute the gradient magnitude and angle arrays,
  M(x,y) and a(x,y) of the input image.

• Form a binary image g, whose value at any pair of
  coordinates (x,y) is given by
• where TM is a threshold used for edge detection,
  A is a specified angle direction, and TA defines
  the band of acceptable directions about A.

51
Edge linking algorithm

• Scan the rows of g and fill (set to 1) all gaps
  (sets of 0s) in each row that do not exceed a
  specified length K.
• By definition, a gap is bounded at both ends by
  one or more 1s.
• Step 3 of the algorithm links edges in the
  horizontal direction.
• To link edges in any other direction ?, we rotate
  g by ? and apply Steps 2-3. We then rotate the
  result back by -?.

52
Local edge linking algorithm 2
Gradient magnitude
Horizontal edge linking
Vertical edge linking
For the horizontal edge linking, TM was set at
30 of the maximum gradient value, A90 because
the gradient direction is vertical to the edge,
TA45. K25, that is, we fill gaps of 25 or
fewer pixels.
53
Edge linking algorithm
The logical OR of the horizontal and vertical
edge linking
The final result after morphological image
post-processing