Digital Signal Processing

Transcription

1 1 PART 1: SUMMARY OF THEORY Digital Signal Processing Image Processing: Introduction to Morphological Processing 1 Part 1: Summary of Theory 1.1 Image (digital) Concept and types An image I is a function D L. The 2D (two-dimensional) image and 3D (three-dimensional) image domains are D Z 2 and D Z 3, respectively. The digital image domain is said to be composed of pixels (for 2D images) or voxels (for 3D images). Note: remember that adequate sampling should be used in continuous signal discretization in order to prevent aliasing. Some types of 2D images are listed below, where the domain D Z 2 and each pixel is associated with two coordinates. Binary image The image set L is {0, 1} (or also {0, 255}), i.e., the value associated with a pixel is 0 or 1. A binary image is an equivalent (isomorphic) concept to set. A binary image would be a membership function of the associated set. Greyscale image The image set L is a subset of the set of integers (L Z). Normally bit depth values of 8 are used, for which reason L is usually {0, 1,..., 255}. Multi-band image In this case L is a Cartesian product of value sets. Each pixel is associated with a tuple of values. Colour images and multi-spectral satellite images are examples of this type of images. José Crespo (Facultad de Informática, UPM). 1.1

2 1.1 Image (digital) 1 PART 1: SUMMARY OF THEORY For example, in the case of images with a colour depth of 8 bits, L is equal to {0, 1,..., 255} {0, 1,..., 255} {0, 1,..., 255}. Other types of images: 3D (greyscale) image 3D or volumetric images have a volumetric domain, i.e., the domain (D Z 3 ) is composed of volumetric elements (voxels) with three coordinates. Examples: medical CT (computerized tomography), MR (magnetic resonance), PET ( positron emission tomography ), SPECT ( Single Photon Emission Computed Tomography ) images. Video image A video image is a 2D image that varies over time (which can be considered as a third dimension). Temporal volumetric image (3D video signal) A temporal volumetric image is a 3D (volumetric) that varies over time. Example: temporal 3D medical image of the heart. We will focus on binary images (or, equivalently, sets) and greyscale images Connectivity The image should have a defined connectivity. Connectivity will be associated with the concept of neighbourhood. There are two common connectivities: 8-connected neighbourhood: Where a pixel (i, j) has eight neighbours: (i 1, j 1), (i 1, j), (i 1, j + 1), (i, j 1), i, j), (i, j + 1), (i + 1, j 1), (i + 1, j), (i + 1, j + 1). 4-connected neighbourhood: where a pixel (i, j) has 4 neighbours: (i 1, j), (i, j 1), (i, j +1), (i+ 1, j). (a) 8-connected neighbourhood (b) 4-connected neighbourhood José Crespo (Facultad de Informática, UPM). 1.2

3 1 PART 1: SUMMARY OF THEORY 1.2 General mathematical fundaments 1.2 General mathematical fundaments The foundations of morphological processing differ from linear signal/image processing. In most cases, images are a very different kettle of fish to other signal types, like voice signals. Specifically, major differences are to be found regarding: The superposition principle The frequency interpretation of linear filters Unlike voice signals, there is no superposition (linear, with addition) of signals in images generally, nor is the frequency location of information good. These differences mean that, when it comes to processing shapes, linear processing has a number of weaknesses. So-called morphological processing is a good alternative for processing the shapes of the structures in an image. It applies a set-theory approach where shapes can be considered satisfactorily. (a) Input image I (b) Example of linear filtering (low-pass filter) (c) Example of morphological filtering (d) Example of morphological filtering 2 José Crespo (Facultad de Informática, UPM). 1.3

4 1.3 Set or binary image case 1 PART 1: SUMMARY OF THEORY The underlying mathematical structure of morphological processing, the lattice, has: An ordering relation Properties: a a (reflexive property). If a b and b a, then a = b (antisymmetric property). If a b and b c, then a c (transitive property). Note: an ordering relation can be total or partial. Two basic operations: Sup (or supremum) The sup of two elements a and b of a lattice L calculates the least element of L that is greater than both a and b. Inf (or infimum) The inf of two elements a and b of a lattice calculates the greatest element of L that is less than both a and b. We will see that these sup and inf operations translate to simple operations in the cases that we will examine. 1.3 Set or binary image case The sets in a space E are elements of the power set P(E), which is the set of all the subsets of E 1. Ordering relation : is the relation of set inclusion. Sup operation : is the operation of set union. 1 I.e., P(E) = {A : A E}. A B = A B José Crespo (Facultad de Informática, UPM). 1.4

5 1 PART 1: SUMMARY OF THEORY 1.4 Greyscale images case Inf operation : is the operation of set intersection. A B = A B Binary sets and images (functions) are equivalent concepts (isomorphs). A binary image can be considered as the membership function of a 2D set. 1.4 Greyscale images case Greyscale images f : E L, where L is a subset of the set of integers Z (like, for example, the range {0, 1,..., 255}), is also associated with a lattice structure. Ordering relation : The ordering relation is derived from the relation existing in Z: f g f(x) Z g(x), x E. Sup operation : (f g)(x) = f(x) Z g(x) = max(f(x), g(x)) Z Inf operation : (f g)(x) = f(x) Z g(x) = min(f(x), g(x)) Z f(x) g(x) x (a) Functions f(x) and g(x) José Crespo (Facultad de Informática, UPM). 1.5

6 1.5 Properties and definitions 1 PART 1: SUMMARY OF THEORY f(x) g(x) f(x) g(x) x x (b) f(x) g(x) (c) f(x) g(x) 1.5 Properties and definitions An increasing transformation ψ defined on a lattice L satisfies that, for all a, b elements of L: a b ψ(a) ψ(b), (1) i.e., if the inputs are ordered, then the respective outputs are also ordered. If ψ is an increasing operator, then we have the following inequalities: ψ(i I ) ψ(i) ψ(i ). (2) ψ(i I ) ψ(i) ψ(i ). (3) if ψ and ψ are increasing operators, then ψ ψ, ψ ψ, and ψψ are all increasing operators. I.e., all compositions of increasing operators, with which we are concerned, are increasing. If I is an input image, an image operator Ψ is extensive if and only if I Ψ(I), (4) i.e., if the output is always greater than or equal to the input. José Crespo (Facultad de Informática, UPM). 1.6

7 2 STRUCTURING ELEMENTS An operator Ψ is anti-extensive if and only if i.e., if the output is always less than or equal to the input. An operator Ψ is idempotent if and only if Ψ(I) I, (5) Ψ(I) = ΨΨ(I), (6) i.e., when Ψ is sequentially applied twice and it leaves the first output unchanged. Two operators Ψ and Ω are dual (of each other) if Ψ = Ω. (7) Clearly, we have also that Ω = Ψ. The complement operation computes the complement of an input. Given an input set A in a space E, then (A) = E \ A, where \ denotes set subtraction. In the case of greyscale images I : D L, a related image inversion operation can be defined. This operation inverts an image reversing the intensity values with respect to the middle point of the intensity value range. For example, for 8-bit images, such an operation would transform a value of 0 into 255, a value of 1 into 254, etc. 2 Structuring elements A structuring element is an auxiliary form (set) used by several operators. This auxiliary set is usually quite a lot smaller in size than the input set. It can be likened to the windows commonly used as a processing aid in linear processing. (Note: there is another, quite infrequently used, type of structuring elements called volumetric elements that are functions rather than sets.) Given a structuring element B, its tranpose, denoted by B, is the inversion of B, with respect to the set of coordinates (0, 0): if (x, y) denotes a point (or pixel) that is a member of B, then B is the set of points (or pixels) such that B = { ( x, y) (x, y) B }. José Crespo (Facultad de Informática, UPM). 1.7

8 3 EROSIONS AND DILATIONS 3 Erosions and dilations Erosions and dilations are the most elementary morphological operators. 3.1 General concepts Generally speaking, erosions and dilations can be said, algebraically (although this is not an operational definition), to materialize as operators that are interchangeable with the sup and inf lattice operations: Dilations δ are increasing operations that satisfy: δ(i I ) = δ(i) δ(i ). (8) Respectively, erosions ε are increasing operations that satisfy: ε(i I ) = ε(i) ε(i ). (9) Erosions and dilations that use structuring elements to transform an input image are a very useful and important type of erosions and dilations. In the following we will first tackle operations from a more straightforward, set theory viewpoint to observe the effect of operations on forms and then we will deal with the expressions associated with the images. 3.2 Erosions ε B and dilations δ B with a structuring element If A denotes an input set and B stands for a structuring element, the erosion with structuring element ε B (A) is located at points x such that B x (the structuring element shifted to x) is completely inside the input set A: ε B (A) = { x B x A } (10) where B x stands for the structuring element B shifted to point (or pixel) x. Note: ε B (A) ε A (B). If A is an input set, the dilation with structuring element δ B is located at points x such that B x (the structuring element shifted to x) has a non-empty intersection (that is touches ) the input set A. δ B (A) = { x B x A } (11) José Crespo (Facultad de Informática, UPM). 1.8

9 3 EROSIONS AND DILATIONS 3.2 Erosions ε B and dilations δ B with a structuring element Note: the source of the dilations and erosions with structuring elements are the so-called Minkowski operations (the Minkowski addition and subtraction). B A (a) Input set A and structuring element B (dark). ε B(A) (b) ε B (A) José Crespo (Facultad de Informática, UPM). 1.9

10 3.2 Erosions ε B and dilations δ B with a structuring element 3 EROSIONS AND DILATIONS B A (a) Input set A and structuring element B (dark). δ B(A) (b) δ B (A) José Crespo (Facultad de Informática, UPM). 1.10

11 3 EROSIONS AND DILATIONS 3.2 Erosions ε B and dilations δ B with a structuring element Some properties of ε B and δ B : Dilation δ B is commutative, i.e., if A denotes an input set, then δ B (A) = δ A (B). If the structuring element B contains the origin of the coordinates, then δ B is extensive, i.e., δ B (I) I, for all images I. Otherwise, the dilations δ B are not generally extensive, although it is always holds that a shift in the input is less than or equal to the output δ B (I). Dilation with a structuring element is associative, i.e., if B is the result of δ C (D) (or δ D (C)), then δ B (I)) = δ C (δ D (I)) = δ D (δ C (I)). Note: e.g., the B structuring element B can be expressed as a dilation of the following structuring elements C and D (B is equal to δ C (D), or to δ D (C)): C D If the centre of coordinates is a member of the structuring element B, then ε B is anti-extensive, i.e., ε B (I) I, for all images I. If the origin of the coordinates is not a member of the structuring element B, then the erosions are not anti-extensive; however, it always holds that ε B (I) is less than or equal to a shift in the input. Erosion with a structuring element is associative, i.e., if B is the result of δ C (D) (or δ D (C)), then ε B (I)) = ε C (ε D (I)) = ε D (ε C (I)). Other expressions for dilations and erosions with a structuring element are as follows: Erosions: ε B (A) = b B A b = b B A b (12) ε B (I) = b B I b (13) José Crespo (Facultad de Informática, UPM). 1.11

12 3.2 Erosions ε B and dilations δ B with a structuring element 3 EROSIONS AND DILATIONS (ε B (I))(x) = min{i(x + b)} (14) b B Dilations: δ B (A) = b B A b = b B A b (15) δ B (I) = b B I b (16) (δ B (I))(x) = max{i(x + b)} (17) b B Expressions (14) and (17) are the expressions that are usually used in practice to calculate dilations and erosions with a structuring element, in the case of both binary and greyscale images. Thus, for example, if the structuring element B is a square of size 3 3, B: square 3 3 the result of the erosion ε B for a particular pixel (i, j) is (using a two-coordinate notation): 2 (ε B (I))(i, j) = min Similarly, for the δ B dilation: (δ B (I))(i, j) = max I(i 1, j 1), I(i 1, j), I(i 1, j + 1), I(i, j 1), I(i, j), I(i, j + 1), I(i + 1, j 1), I(i + 1, j), I(i + 1, j + 1) I(i 1, j 1), I(i 1, j), I(i 1, j + 1), I(i, j 1), I(i, j), I(i, j + 1), I(i + 1, j 1), I(i + 1, j), I(i + 1, j + 1) 2 Note: Care should, of course, be taken not to exceed the image boundaries. José Crespo (Facultad de Informática, UPM). 1.12

13 4 OPENINGS AND CLOSINGS 4 Openings and closings 4.1 General concepts A morphological filter is an increasing and idempotent transformation. (Note that the word filter has a different meaning here than in linear processing, where, in practice, it signifies any transformation.) The transformations in section 3, erosions and dilations, are not idempotent (and, therefore, are not morphological filters). Let us now look at other more complex transformations that do have the property of idempotency. An opening γ is an anti-extensive morphological filter. A closing ϕ is an extensive morphological filter. As in the case of erosions and dilations, openings and closings that use a structuring element are an important type of openings and closings. 4.2 Openings and closings with a structuring element An opening with a structuring element B of an input set A, denoted by γ B (A), is the set of points x that are members of a shift of B that is completely included in the input set A. It is calculated by sequentially combining an erosion and dilation with a structuring element: γ B = δ Bε B, The closing with structuring element ϕ B can be defined by applying the duality with the opening: the closing with the structuring element of an input set A, denoted by ϕ B (A) is the complement of the points that are members of the shift of B that is included in the complement of A ( (A)). In other words, ϕ B (A) is the complement of the opening γ B of (A). It is calculated as: ϕ B = ε Bδ B, The effects of these filters are (expressed in the binary framework): The opening γ B removes the light structures that are smaller than the structuring element B. This way, it will remove connected components that are less than B. José Crespo (Facultad de Informática, UPM). 1.13

14 5 OTHER TYPES OF FILTERS The closing ϕ B will fill in the dark structures that are smaller than the structuring element B. Therefore, it will close up holes in the input set that are less than B. Some examples Directly applying the definition of inclusion: (a) Input set A (dark) (b) B (c) γ B (A) (d) B (e) γ B (A) (f) B (g) γ B (A) Applying elementary erosion and dilation operations: (a) Input set A (dark) (b) B (c) ε B (A) (d) γ B (A) = δ Bε B (A) 5 Other types of filters (e) B (f) ε B (A) (g) γ B (A) = δ Bε B (A) Generally, the effects of opening and closing are usually combined by applying the two sequentially, one after the other. They are called alternate filters: ϕγ γϕ José Crespo (Facultad de Informática, UPM). 1.14

15 6 GRADIENT AND TOP-HAT TRANSFORM The alternate filters ϕγ and γϕ are idempotent, i.e., they are morphological filters. There is no ordering relation between ϕγ and γϕ, or with the input: ϕγ γϕ ϕγ id ϕγ id id γϕ id 6 Gradient and top-hat transform Morphological gradient The morphological gradient operator is an approximation of the module of the image derivative. It is a measure of the change of the function. δ B (I) ε B (I) (18) (Note: the symbol - denotes pixelwise integer subtraction.) The gradient of expression (18) is usually called the Beucher gradient. In binary images, the morphological gradient extracts the boundary of the forms, specifically the internal and external boundary. To extract either the internal boundary or the external of a binary image, we calculate I ε B (I) or δ B (I) I, respectively. (Note: the morphological gradient is an example of a residual operation.) Top-hat transforms Top-hat transforms are used to extract structures of interest. They are attractive transforms for this purpose, because, for one thing, it is often easier to use a filter to remove all, or part of, the structures of interest than to delete the irrelevant parts of an image. Additionally, these transforms are quite robust to illumination changes. There are three types of transforms depending on whether we are interested in light, dark or both structures: White top-hat (or top-hat for light structures) José Crespo (Facultad de Informática, UPM). 1.15

16 7 GRANULOMETRIES AND ANTI-GRANULOMETRIES Black top-hat (or top-hat for dark structures) I γ(i) ϕ(i) I Symmetric top-hat (or top-hat for light and dark structures) ϕ(i) γ(i) The procedure is to first calculate a top-hat and then apply a threshold to output a binary image that marks (or indicates) the structures of interest. 7 Granulometries and anti-granulometries 7.1 Definitions Axioms of granulometries or size distributions: Increasingness Anti-extensivity Absorption: If Ψ i, Ψ j are two transforms of a size distribution, where i j, then Ψ i Ψ j = Ψ j Ψ i = Ψ max(i,j) = Ψ j. (19) A family of openings {γ i }, where i S = {1,..., n}, is a granulometry if, for all i, j S, i j γ i γ J. (20) I.e., the openings are ordered. The dual concept is anti-granulometry. A family of closings {ϕ i }, where i S = {1,..., n}, is an antigranulometry if, for all i, j S, José Crespo (Facultad de Informática, UPM). 1.16

17 7 GRANULOMETRIES AND ANTI-GRANULOMETRIES 7.2 Granulometries and anti-granulometries with a structuring element i j ϕ i ϕ J. (21) I.e., the closings are ordered. In practice, the term granulometry often covers the use of openings and closings (i.e., including both a granulometry and anti-granulometry component). For example, a granulometry of indices [ 3, 3] would use the filters: γ 3, γ 2, γ 1, id, ϕ 1, ϕ 2, ϕ 3. The symbol id denotes the identity operator that does not modify the input. 7.2 Granulometries and anti-granulometries with a structuring element The granulometries and anti-granulometries with structuring elements use families of structuring elements that guarantee the ordering relation between openings and closings. The family of square structuring elements below would be adequate. It uses the usual size convention that the index i translates to a square of size 2i + 1 (or, in the case of circles, to a circle of radius i). (a) i = 0 (b) i = 1 (c) i = 2 (d) i = 3 The following condition must hold: γ (i 1)B (ib) = ib, for i 1. (22) 7.3 Granulometric curve Generally, a criterion or measure of the output of the granulometry and/or anti-granulometry filters is applied. The results are then used to build what is called a granulometric curve, which, sometimes and depending on the choice of structuring element, is useful for describing and analysing the forms of the present structures. That criterion or measure must be increasing, which means that the curve will be monotonic increasing (if x y, then f(x) f(y)) if part of the openings are associated with negative indices, as is usually the case. The figures below illustrate two cases, a binary and a greyscale example. Note that the granulometric curves are increasing from left to right. Often, the derivative of the granulometric curve is also calculated to better reflect the changes. José Crespo (Facultad de Informática, UPM). 1.17

18 7.3 Granulometric curve 7 GRANULOMETRIES AND ANTI-GRANULOMETRIES (a) Input set A (white) (b) γ 4 (A) (c) γ 6 (A) (d) γ 8 (A) (e) Granulometric curve (range: [ 15, 15]) Figure 1: Granulometry: binary image example. José Crespo (Facultad de Informática, UPM). 1.18

19 7 GRANULOMETRIES AND ANTI-GRANULOMETRIES 7.3 Granulometric curve (a) Input image I (b) γ 4 (I) (c) γ 6 (I) (d) γ 8 (I) (e) Granulometric curve (range: [ 15, 15]) Figure 2: Granulometry: (non-binary) image example. José Crespo (Facultad de Informática, UPM). 1.19

20 8 HIT-OR-MISS TRANSFORM 8 Hit-or-miss transform The hit-or-miss transformation is an operator that uses a composite structuring element, i.e., uses two structuring elements B and C. The HMT (B,C) of an input set A calculates the set of points x such that the shifted structuring elements B x and C x are included in, respectively, A and (A): HMT (B,C) (A) = {x B x A, C x (A)} Note that B and C should be disjoint (otherwise, the output would be empty). We have that: HMT (B,C) (A) = ε B (A) ε C ( (A)) For example, the following composite structuring element would extract the upper vertical endpoints: B (dark) and C (white) Note: as the structuring elements B and C are disjoint, the above representation visualizes both structuring elements in different colours in the same figure. The composite structuring element below would extract the lower vertical endpoints: B (dark) and C (white) Therefore, we can extract (upper and lower) vertical endpoints of an input set A as: HMT (B,C) (A) HMT (B,C )(A). (Note: in the above two examples, the centre belongs to the first structuring element. however, the centre could belong to the second structuring element in other cases.) José Crespo (Facultad de Informática, UPM). 1.20

21 9 PART 2: PRACTICAL ASSIGNMENT 9 Part 2: Practical Assignment Image Processing: Introduction to Morphological Processing SURNAME: SURNAME: NAME: NAME: Exercise 1 Erosions and Dilations Let I be the input image of the file squares01.pgm. Let B be a square structuring element with sides measuring 3. Is ε B (I) I? Is ε B (I) < I? I Let I be the input image of file squares02.pgm. Let B be a square structuring element with sides measuring 3. Is δ B (I) I? Is δ B (I) > I? I Exercise 2 Openings and structuring elements If there is more than one structuring element that meets the specifications in this exercise, choose the smallest sized one (and without holes). José Crespo (Facultad de Informática, UPM). 1.21

22 9 PART 2: PRACTICAL ASSIGNMENT Consider image I in file fig01 32.pgm. What structuring element B should you use to output the results shown in the image in fig02 32.pgm for the input image I with an opening γ B? I γ B (I) Consider image I in file fig01 32.pgm. What structuring element B should you use to output the results shown in the image in fig03 32.pgm for the input image I with an opening γ B? I γ B (I) Is there any structuring element B greater than a pixel such that γ B does not modify the image in file fig04 32.pgm, and, if so, which would it be? I γ B (I) José Crespo (Facultad de Informática, UPM). 1.22

23 9 PART 2: PRACTICAL ASSIGNMENT Exercise 3 Noise suppression Let I be the input image in file isn 256.pgm, which has added binary impulse noise ( salt-and-pepper noise). I Let B be a square structuring element of side 3. Calculate: γ B (I), ϕ B (I), ϕ B γ B (I), γ B ϕ B (I). Of the above four filters, state which are the best two for noise suppression of I. Exercise 4 Hit-or-miss transforms Let the input image I in file hm01 32.pgm be: I Suppose we want to use two hit-or-miss transforms to detect, respectively, the isolated points (there are 3 pixels in the figure) and the centre point of the cross intersection (1 pixel). Design two composite structuring elements suitable for the transforms, check the results in Matlab and write them in the box. Exercise 5 Decomposition of structuring elements Let the structuring elements be as follows: José Crespo (Facultad de Informática, UPM). 1.23

24 9 PART 2: PRACTICAL ASSIGNMENT Note that B is the result of δ C (D) (or δ D (C)). B C D Using the greyscale image cam 74.pgm as input, check that δ B (I)) = δ C (δ D (I)). State the number of max operations to be performed to process the square image of size N N for both alternatives (δ B (I)) or δ C (δ D (I))). Suppose that the sizes of B, C and D are, respectively, M M, 1 M and M 1. (Note: ignore any edge effects, that is, process all the image pixels in the same way.) OPTIONAL PART Exercise 6 Composition of openings In the following exercise we will experiment with compositions of openings. A result of morphological filter theory is that the sup of openings is likewise an opening. Let I be the file input image opc01 32.pgm. Let B be a 1 3 horizontal structuring element, and C a 3 1 vertical structuring element: I B C (1) Get the result: I = γ B (I) γ C (I). (2) Then calculate: I = γ B (I ) γ C (I ). José Crespo (Facultad de Informática, UPM). 1.24

25 9 PART 2: PRACTICAL ASSIGNMENT Is I = I? Likewise, after observing the output, write two or three lines discussing what effect the operation (γ B γc )(I) has. Exercise 7 Granulometries In this exercise, use the image of file gran01 64.pgm as the input image. I Calculate the volumes (saved as a vector) of γ i (I), 0 i 5, where γ i denotes an opening with a square structuring element of side 2i + 1. γ 0 (I) is assumed to be I. Use plot to visualize the vector of volumes, and plot the resulting graph, specifying the associated volumes that you have calculated. Note: remember that the openings are associated with negative indices on the granulometric curve. Exercise 8 Increasing operators An increasing operator ψ on sets satisfies ψ(a) ψ(b) if A B, for any pair of sets A, B. Let C, D be two sets. If we know that ψ(d) is (empty set), and that C D, can we find out what the result of ψ(c) is? And, if so, what would it be? José Crespo (Facultad de Informática, UPM). 1.25

26 9 PART 2: PRACTICAL ASSIGNMENT Appendix Simple auxiliary functions related to orderings and the sup, inf and image inversion operations: immareequal.m immareequal (im1, im2): si im1 == im2. immislessthan.m immislessthan (im1, im2): si im1 im2. immislessthanstrict.m immislessthanstrict (im1, im2): si im1 < im2. immislargerthan.m immislargerthan (im1, im2): si im1 im2. immislargerthanstrict.m immislargerthanstrict (im1, im2): si im1 > im2. imminf.m imminf (im1, im2): im1 im2. immsup.m immsup (im1, im2): im1 im2. immvolume.m immvolume (im): volume (cumulative sum of the pixel values) of im. There is also a function for visualizing an image through previous upsampling: factor). Some Matlab functions related to the practical assignment: immshowu (im, José Crespo (Facultad de Informática, UPM). 1.26

27 9 PART 2: PRACTICAL ASSIGNMENT imread: reads images. imwrite: writes images. imshow: shows images. imerode: performs erosion with a structuring element. imdilate: performs dilation with a structuring element. imopen: performs opening with a structuring element. imclose: performs closing with a structuring element. imcomplement: performs the complement / image inversion operation. strel: creates structuring elements of different types. Square of side 3: st = strel( square,3) Disk of radius 1: st = strel( disk,1) Diamond of radius 1: st = strel( diamond,1) Horizontal line of length 3: st = strel ( line, 3, 0); Vertical line of length 3: st = strel ( line, 3, 90); Arbitrary element: starray = [1 0 1; 0 1 0; 1 0 1]; st = strel( arbitrary, starray); José Crespo (Facultad de Informática, UPM). 1.27