Introduction to Medical Image Analysis

Transcription

1 Introduction to Medical Image Analysis Rasmus R. Paulsen DTU Compute

2 Lecture 7 Pixel Classification 9.00 Lecture Exercises Lunch break Exercises 2 DTU Compute, Technical University of Denmark

3 What can you do after today? Describe the concept of pixel classification Use Matlab to select pixel training data Compute the pixel value ranges in a minimum distance classifier Implement and use a minimum distance classifier Describe how a pixel value histogram can be approximated using a Gaussian distribution Describe how the pixel value ranges can be selected in a parametric classifier Implement and use a parametric classifier Decide if a minimum distance or a parametric classifier is appropriate based on the training data Explain the concept of Bayesian classification 3 DTU Compute, Technical University of Denmark

4 Classification Take a measurement and put it into a class Measurement Classes? Wheels: 2 HP: 50 Weight: 200 Classifier Bike Truck Car Motorbike Train Bus 4 DTU Compute, Technical University of Denmark

5 General Classification Multi-dimensional measurement Pre-defined classes Can also be found automatically can be very difficult! 5 DTU Compute, Technical University of Denmark

6 Pixel Classification CT scan of human head Classify each pixel Independent of neighbours Also called labelling Put a label on each pixel We look at the pixel value and assign them a label Labels already defined Background Soft-Tissue Trabecular Bone Hard Bone 6 DTU Compute, Technical University of Denmark

7 Quiz: Two class pixel classification? Background and object A) Median filter B) Threshold 15 C) Brightness D) Morphological Erosion E) BLOB analysis A B C D E 7 DTU Compute, Technical University of Denmark

8 Pixel Classification formal definition Pixel value (the measurement) v R k classes C = c 1,, c k Classification rule c: R {c 1,, c k } 8 DTU Compute, Technical University of Denmark

9 Pixel Classification example Pixel value v [0,255] Set of 4 classes C={background, soft-tissue, trabeculae, bone} Classification rule c: v {background, soft tissue, trabeculae, bone} How do we construct a classification rule? 9 DTU Compute, Technical University of Denmark

10 Pixel classification rule c: v {background, soft tissue, trabeculae, bone} background trabeculae soft-tissue bone How do we do this? 10 DTU Compute, Technical University of Denmark

11 Pixel classification rule manual inspection c: v {background, soft tissue, trabeculae, bone} Looking at some few pixels background soft-tissue trabeculae bone 11 DTU Compute, Technical University of Denmark

12 Pixel classification rule manual inspection c: v {background, soft tissue, trabeculae, bone} Looking at some few pixels New pixel where do we put it? background soft-tissue trabeculae bone 12 DTU Compute, Technical University of Denmark

13 Pixel classification rule manual inspection c: v {background, soft tissue, trabeculae, bone} Looking at some few pixels New pixel where do we put it? Measure the distance to the other classes Select the closest class d b d st Minimum distance classification background soft-tissue trabeculae bone 13 DTU Compute, Technical University of Denmark

14 Pixel classification rule Minimum Distance Classification The possible pixel values are divided into ranges Here the distance to background is equal to soft-tissue Background range soft-tissue range Trabecular range Bone range background soft-tissue trabeculae bone 14 DTU Compute, Technical University of Denmark

15 Pixel classification rule Minimum Distance Classification background, if v (4 + 67)/2 soft tissue, if < v 2 2 c v = trabeculae, if < v bone, if v > 2 Background soft-tissue Trabecular Bone range range range range background soft-tissue trabeculae bone 15 DTU Compute, Technical University of Denmark

16 Pixel classification rule For all pixel in the image do c v = background, if v (4 + 67)/2 soft tissue, if < v trabeculae, if < v bone, if v > 2 16 DTU Compute, Technical University of Denmark

17 Pixel Classification example Background CT scan of human head Soft-Tissue Trabecular Bone Hard Bone 17 DTU Compute, Technical University of Denmark

18 Better range selection Guessing range values is not a good idea Better to use training data Start by selecting representative regions from an image Annotation To mark points, regions, lines or other significant structures 18 DTU Compute, Technical University of Denmark

19 Classifier training - annotation An expert is asked how many different tissue types that are possible Then the expert is asked to mark representative regions of the selected tissue types Background Soft-Tissue Trabecular Bone Hard Bone 19 DTU Compute, Technical University of Denmark

20 Classifier training region selection Many tools exist Matlab tool roipoly Select closed regions using a piecewise polygon Training is only done once! Optimally, the training can be used on many pictures that contains the same tissue types 20 DTU Compute, Technical University of Denmark

21 Initial analysis - histograms Gaussian 21 DTU Compute, Technical University of Denmark

22 Initial analysis - histograms Class separation 22 DTU Compute, Technical University of Denmark

23 Simple pixel statistics Calculate the average (mean) and the standard deviation of each class Standard deviation Average 23 DTU Compute, Technical University of Denmark

24 24 DTU Compute, Technical University of Denmark

25 Minimum distance classification Any objections? The pixel value ranges are not always in good correspondence with the histograms? 25 DTU Compute, Technical University of Denmark

26 Minimum distance classification A) Baggrund B) Blødt væv C) Fedt D) Knogle E) Ingen af dem A B C D E 26 DTU Compute, Technical University of Denmark

27 Parametric classification Describe the histogram using a few parameters Gaussian/Normal distribution Average μ Standard deviation σ f x = 1 σ 2π exp (x μ)2 2σ 2 Trabecular bone Only two values needed 27 DTU Compute, Technical University of Denmark

28 Parametric classification Training pixel values Estimated average Estimated standard deviation Trabecular bone f x = 1 σ 2π exp (x μ)2 2σ 2 28 DTU Compute, Technical University of Denmark

29 Parametric classification Fit a Gaussian to the training pixels for all classes What do we see here? What is the difference between the two classes? Trabeculae has much higher variation in the pixel values 29 DTU Compute, Technical University of Denmark

30 Two tissue types minimum distance v = 78 A) Trabeculae B) Soft-tissue 12 5 v = 78 A B Minimum distance classifier 30 DTU Compute, Technical University of Denmark

31 Parametric classification New pixel with value 78 Is it soft-tissue or trabecular bone? Minimum distance classifier? Soft-tissue Is that fair? Soft-tissue Gaussian says Extremely low probability that this pixel is soft-tissue v = DTU Compute, Technical University of Denmark

32 Two tissue types parametric classification A) Trabeculae B) Soft-tissue 14 1 A B 32 DTU Compute, Technical University of Denmark v = 78

33 Parametric classification repeat the question New pixel with value 78 Is it soft-tissue or trabecular bone? Most probably trabecular bone Where should we set the limit? Where the two Gaussians cross! v = DTU Compute, Technical University of Denmark

34 Parametric classification ranges The pixel value ranges depends on The average The standard deviation Compared to the minimum distance classifier Only the average Soft-tissue Trabecular bone 34 DTU Compute, Technical University of Denmark

35 Parametric classification how to Select training pixels for each class Fit Gaussians to each class Use Gaussians to determine pixel value ranges Little bit difficult with the Gaussians 35 DTU Compute, Technical University of Denmark

36 Parametric classifier - ranges f 1 (x) > f 2 (x) We want to compute where they cross f 1 (x) < f 2 (x) Create a lookup table: Run through all 256 possible pixel values Check which Gaussian is the highest Store the [value, class] in the table 36 DTU Compute, Technical University of Denmark

37 Alternatively analytic solution The two Gaussians Intercept at 37 DTU Compute, Technical University of Denmark

38 Class ranges A) [0,45], ]45, 75], ]75,255] B) [40,60], ]60,100],]100,140] C) [0, 60],]60,80],]80,255] D) [0,60],]60,100],]100,255] E) [0,75],[75,100],]100,255] A B C D E 38 DTU Compute, Technical University of Denmark

39 Parametric classification A) Baggrund B) Blødt væv C) Nyre D) Milt E) Knogle A B C D E 39 DTU Compute, Technical University of Denmark

40 Kurset indtil videre A) Jeg har ikke lært noget B) Jeg har ikke lært ret meget C) Det er ok D) Jeg har lært en del E) Jeg har lært meget A B C D E 40 DTU Compute, Technical University of Denmark

41 Undervisning hastighed i undervisning A) Kom nuuuu! Det går alt for langsomt B) Jeg kan sagtens følge med og strikke sweater samtidigt C) Det er fint tempo D) Jeg skal virkelig koncentrere mig for at følge med E) Stop! Vent! Alt for hurtigt A B C D E 41 DTU Compute, Technical University of Denmark

42 Thomas Bayes English mathematician and Presbyterian minister Bayes theorem P A B = P B A P(A) P(B) Wikipedia 42 DTU Compute, Technical University of Denmark

43 Bayesian Classification Pure parametric classifier assumes equal amount of different tissue types Area = 1 43 DTU Compute, Technical University of Denmark

44 Bayesian Classification Much more softtissue than trabecular bone Area = 1 How do we handle that? 44 DTU Compute, Technical University of Denmark

45 Bayesian Classification An expert tells us that a CT scan of a head contains 20% Trabecular bone 50% Soft-tissue Picking a random pixel in the image 20% Chance that it is trabecular bone 50% Chance that it is softtissue How do use that? 45 DTU Compute, Technical University of Denmark

46 Bayesian Classification histogram scaling Scaled with 0.50 Scaled with 0.20 Parametric classifier Bayesian classifier 46 DTU Compute, Technical University of Denmark Little change in class border (sometimes significant changes)

47 Formal definition Given a pixel value v What is the probability that the pixel belongs to class C i Example: If the pixel value is 78, what is the probability that the pixel is bone P c i v = P v c i P(c i ) P(v) 47 DTU Compute, Technical University of Denmark

48 Formal definition Constant ignored from now on P c i v = P v c i P(c i ) P(v) 48 DTU Compute, Technical University of Denmark

49 Formal definition The a priori probability (what is known from before) Example: From general biology it is known that 20% of a brain CT scan is trabecular bone. Therefore P(trabecular) = 0.20 P c i v = P v c i P(c i ) P(v) 49 DTU Compute, Technical University of Denmark

50 Formal definition The class conditional probability Given a class, what is the probability of a pixel with value v Example: If we consider class = soft-tissue. What is the probability that the pixel value is 78? Very low P c i v = P v c i P(c i ) P(v) 50 DTU Compute, Technical University of Denmark

51 Formal definition sum up P c i v = P v c i P(c i ) P(v) c i = trabeculae 51 DTU Compute, Technical University of Denmark

52 Bayesian classification how to Select training pixels for each class Fit Gaussians to each class Ask an expert for the prior probabilities (how much there normally is in total of each type) For each pixel in the image Compute P(v c i ) for each class (the a posterior probability) Select the class with the highest P(c i v) P c i v = P v c i P(c i ) P(v) 52 DTU Compute, Technical University of Denmark

53 When to use Bayesian classification The parametric classifier is good when there are approximately the same amount of all type of tissues Use Bayesian classification if there are very little or very much of some types 53 DTU Compute, Technical University of Denmark

54 Multiple choice A) 1 B) 2 C) 3 D) 4 E) DTU Compute, Technical University of Denmark A B C D E

55 Tissue classification A) Baggrund B) Blødt væv C) Lever D) Milt E) Knogle A B C D E 55 DTU Compute, Technical University of Denmark

56 What can you do after today? Describe the concept of pixel classification Use Matlab to select pixel training data Compute the pixel value ranges in a minimum distance classifier Implement and use a minimum distance classifier Describe how a pixel value histogram can be approximated using a Gaussian distribution Describe how the pixel value ranges can be selected in a parametric classifier Implement and use a parametric classifier Decide if a minimum distance or a parametric classifier is appropriate based on the training data Explain the concept of Bayesian classification 56 DTU Compute, Technical University of Denmark

57 Next week X-ray and CT Eksternt foredrag: Algorithm Specialist Mikkel Stegmann, Fingerprint Cards 57 DTU Compute, Technical University of Denmark

58 Exercises? 58 DTU Compute, Technical University of Denmark