52:, knowledge about an object (e.g., an image) depends on the manner in which we probe (observe) it. In particular, the choice of a certain structuring element for a particular morphological operation influences the information one can obtain. There are two main characteristics that are directly related to structuring elements:
226:
When used by a hit-or-miss transform, usually the structuring element is a composite of two disjoint sets (two simple structuring elements), one associated to the foreground, and one associated to the background of the image to be probed. In this case, an alternative representation of the composite
59:. For example, the structuring element can be a "ball" or a line; convex or a ring, etc. By choosing a particular structuring element, one sets a way of differentiating some objects (or parts of objects) from others, according to their shape or spatial orientation.
24:
is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image. It is typically used in morphological operations, such as
117:
square. Setting the size of the structuring element is similar to setting the observation scale, and setting the criterion to differentiate image objects or features according to size.
115:
89:
294:
278:
310:
126:
Structuring elements are particular cases of binary images, usually being small and simple. In mathematical morphology,
17:
42:
26:
231:
which are either set (1, associated to the foreground), not set (0, associated to the background) or "
220:
38:
34:
30:
94:
68:
290:
274:
223:, assuming the values 1 (if the pixel belongs to the structuring element) or 0 (otherwise).
49:
135:
304:
232:
127:
215:
In the discrete case, a structuring element can also be represented as a set of
149:. Here are some examples of widely used structuring elements (denoted by
131:
228:
216:
192:={(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}.
97:
71:
287:
Image
Analysis and Mathematical Morphology, Volume 1
109:
83:
271:An Introduction to Morphological Image Processing
65:. For example, one structuring element can be a
8:
96:
70:
244:
122:Mathematical particulars and examples
7:
227:structuring element is as a set of
211:={(-1,0),(0,-1),(0,0),(0,1),(1,0)}.
14:
252:
1:
110:{\displaystyle 21\times 21}
327:
188:is a 3x3 square, that is,
169:is an open disk of radius
207:is the "cross" given by:
173:, centered at the origin.
84:{\displaystyle 3\times 3}
311:Mathematical morphology
18:mathematical morphology
111:
85:
269:Edward R. Dougherty,
255:), chapter 1, page 1.
145:, for some dimension
112:
86:
43:hit-or-miss transform
141:or the integer grid
95:
69:
22:structuring element
107:
81:
41:, as well as the
318:
282:
256:
249:
116:
114:
113:
108:
90:
88:
87:
82:
50:Georges Matheron
326:
325:
321:
320:
319:
317:
316:
315:
301:
300:
268:
265:
260:
259:
250:
246:
241:
136:Euclidean space
124:
93:
92:
67:
66:
12:
11:
5:
324:
322:
314:
313:
303:
302:
299:
298:
283:
264:
261:
258:
257:
253:Dougherty 1992
243:
242:
240:
237:
213:
212:
193:
174:
123:
120:
119:
118:
106:
103:
100:
80:
77:
74:
60:
13:
10:
9:
6:
4:
3:
2:
323:
312:
309:
308:
306:
296:
295:0-12-637241-1
292:
288:
284:
280:
279:0-8194-0845-X
276:
272:
267:
266:
262:
254:
248:
245:
238:
236:
234:
230:
224:
222:
218:
210:
206:
202:
198:
194:
191:
187:
183:
179:
175:
172:
168:
164:
160:
156:
155:
154:
152:
148:
144:
140:
137:
133:
129:
128:binary images
121:
104:
101:
98:
78:
75:
72:
64:
61:
58:
55:
54:
53:
51:
48:According to
46:
44:
40:
36:
32:
28:
23:
19:
286:
285:Jean Serra,
270:
247:
225:
214:
208:
204:
200:
196:
189:
185:
181:
177:
170:
166:
162:
158:
150:
146:
142:
138:
125:
91:square or a
62:
56:
47:
21:
15:
263:References
233:don't care
102:×
76:×
305:Category
27:dilation
132:subsets
39:closing
35:opening
31:erosion
297:(1982)
293:
281:(1992)
277:
229:pixels
217:pixels
37:, and
251:See (
239:Notes
219:on a
134:of a
57:Shape
20:, a
291:ISBN
275:ISBN
221:grid
195:Let
176:Let
157:Let
130:are
63:Size
235:".
153:):
16:In
307::
289:,
273:,
203:;
184:;
165:;
105:21
99:21
45:.
33:,
29:,
209:B
205:B
201:Z
199:=
197:E
190:B
186:B
182:Z
180:=
178:E
171:r
167:B
163:R
161:=
159:E
151:B
147:d
143:Z
139:R
79:3
73:3
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.