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form of the R-table in the transform stage. For every edge point on the test image, the properties of the point are looked up on the R-table and reference point is retrieved and the appropriate cell in a matrix called the
Accumulator matrix is incremented. The cell with maximum 'votes' in the Accumulator matrix can be a possible point of existence of fixed reference of the object in the test image.
458:
is then accomplished by straightforward transformations to this table. The key generalization to arbitrary shapes is the use of directional information. Given any shape and a fixed reference point on it, instead of a parametric curve, the information provided by the boundary pixels is stored in the
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Ballard suggested using orientation information of the edge decreasing the cost of the computation. Many efficient GHT techniques have been suggested such as the SC-GHT (Using slope and curvature as local properties). Davis and Yam also suggested an extension of Merlin's work for orientation and
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The problem of finding the object (described with a model) in an image can be solved by finding the model's position in the image. With the generalized Hough transform, the problem of finding the model's position is transformed to a problem of finding the transformation's parameter that maps the
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for the shape. Thus it is theoretically possible to use the two points in image space to reduce the locus in parameter space to a single point. However, the difficulties of finding the intersection points of the two surfaces in parameter space will make this approach unfeasible for most cases.
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of reference points in the Hough space. Every pixel of the image votes for its corresponding reference points. The maximum points of the Hough space indicate possible reference points of the pattern in the image. This maximum can be found by scanning the Hough space or by solving a
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1060:. The concern with this transform is that the choice of reference can greatly affect the accuracy. To overcome this, Ballard has suggested smoothing the resultant accumulator with a composite smoothing template. The composite smoothing template
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etc.). In these cases, we have knowledge of the shape and aim to find out its location and orientation in the image. This modification enables the Hough transform to be used to detect an arbitrary object described with its model.
729:. These entry points give us each possible position for the reference point. Although some bogus points may be calculated, given that the object exists in the image, a maximum will occur at the reference point. Maxima in
753:. The R-table can also be used to increment this larger dimensional space since different orientations and scales correspond to easily computed transformations of the table. Denote a particular R-table for a shape
535:. Having done this for each point, the R-table will fully represent the template object. Also, since the generation phase is invertible, we may use it to localise object occurrences at other places in the image.
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structure. It results in improved efficiency in finding endpoints of line segments and improved robustness and reliability in extracting lines in noisy situations, at a slightly increased cost of memory.
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A pair of edge pixels can be used to reduce the parameter space. Using the R-table and the properties as described above, each edge pixel defines a surface in the four-dimensional accumulator space of
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761:. Simple transformations to this table will allow it to detect scaled or rotated instances of the same shape. For example, if the shape is scaled by s and this transformation is denoted by
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A. A. Kassim, T. Tan, K. H. Tan, "A comparative study of efficient generalised Hough transform techniques", Image and Vision
Computing, Volume 17, Issue 10, Pages 737-748, August 1999
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scale factors. An algorithm can compute the best set of parameters for a given shape from edge pixel data. These parameters do not have equal status. The reference origin location,
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Merlin and Farber showed how to use a Hough algorithm when the desired curves could not be described analytically. It was a precursor to
Ballard's algorithm that was restricted to
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Observing that the global Hough transform can be obtained by the summation of local Hough transforms of disjoint sub-region, Heather and Yang proposed a method which involves the
811:. Another property which will be useful in describing the composition of generalized Hough transforms is the change of reference point. If we want to choose a new reference point
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The Merlin-Farber algorithm is impractical for real image data as in an image with many edge pixels, it finds many false positives due to repetitive pixel arrangements.
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The original implementation of the GHT used edge information to define a mapping from orientation of an edge point to a reference point of the shape. In the case of a
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For a fixed orientation of shape, the accumulator array was two-dimensional in the reference point co-ordinates. To search for shapes of arbitrary orientation
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scale invariant matching which complement's
Ballard's work but does not include Ballard's utilization of edge-slope information and composite structures
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99:
94:
749:, these two parameters are added to the shape description. The accumulator array now consists of four dimensions corresponding to the parameters
450:, is described in terms of a template table called the R table of possible edge pixel orientations. The computation of the additional parameters
17:
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where pixels can be either black or white, every black pixel of the image can be a black pixel of the desired pattern thus creating a
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It has substantial computational and storage requirements which become acute when object orientation and scale have to be considered.
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model into the image. Given the value of the transformation's parameter, the position of the model in the image can be determined.
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D.H. Ballard, "Generalizing the Hough
Transform to Detect Arbitrary Shapes", Pattern Recognition, Vol.13, No.2, p.111-122, 1981
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To generalize the Hough algorithm to non-analytic curves, Ballard defines the following parameters for a generalized shape:
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It is robust to partial or slightly deformed shapes (i.e., robust to recognition under occlusion).
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1506:{\displaystyle \cos(\pi -\alpha )=x'/r\ \ {\text{or}}\ \ x'=r\cos(\pi -\alpha )=-r\cos(\alpha )}
2385:"Hierarchical Generalized Hough Transforms and Line Segment Based Generalized Hough Transforms"
1639:{\displaystyle \sin(\pi -\alpha )=y'/r\ \ {\text{or}}\ \ y'=r\sin(\pi -\alpha )=r\sin(\alpha )}
392:
1836:(0) Convert the sample shape image into an edge image using any edge detecting algorithm like
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350:. The Hough transform was initially developed to detect analytically defined shapes (e.g.,
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of the image into sub-images, each with their own parameter space, and organized in a
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http://www.mathworks.com/matlabcentral/fileexchange/44166-generalized-hough-transform
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721:(initialized to a maximum size of the image) where r is a table entry indexed by
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http://docs.opencv.org/master/dc/d46/classcv_1_1GeneralizedHoughBallard.html
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It can find multiple occurrences of a shape during the same processing pass.
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for the shape (typically chosen inside the shape). For each boundary point
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FPGA implementation of generalized Hough transforms, IEEE Digital
Library
2183:(3) Possible locations of the object contour are given by local maxima in
1913:(3) Possible locations of the object contour are given by local maxima in
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It is robust to the presence of additional structures in the image.
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2400:, The 2nd Canadian Conference on Computer and Robot Vision, 2005.
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http://www.itriacasa.it/generalized-hough-transform/default.html
1772:(0) Convert the sample shape image into an edge image using any
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If the shape S has a composite structure consisting of subparts
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2432:"A generalized hough-like transformation for shape recognition"
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1053:{\displaystyle R_{\phi }=T_{s}\left\{T_{\theta }\left\right\}}
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http://www.irit.fr/~Julien.Pinquier/Docs/Hough_transform.html
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Tutorial and implementation of generalized Hough transforms
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Geometry of shape detection for generalized Hough transform
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2434:. University of Texas Computer Sciences, TR-134, Feb 1980.
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1797:(2) Draw a line from the reference point to the boundary
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2339:"A Parallel Mechanism for Detecting Curves in Pictures"
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of individual smoothing templates of the sub-shapes.
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or as the radial distance and the angle between them
2326:. In Proceedings of Sysid 2009, Saint-Malo, France.
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1146:{\displaystyle H(y)=\sum _{i=1}^{N}h_{i}(y-y_{i})}
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2387:, University of Texas Computer Sciences, Nov 1980
1193:The implementation uses the following equations:
819:then the modification to the R-table is given by
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1168:corresponds to possible instances of the shape.
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1946:Suppose the object has undergone some rotation
733:correspond to possible instances of the shape.
383:, each of them corresponding to a black pixel.
2477:implementation of generalized Hough transform
2450:implementation of generalized Hough transform
2398:"Spatial Decomposition of the Hough Transform"
2316:
1370:{\displaystyle y=y_{c}+y'\ \ or\ \ y_{c}=y-y'}
1278:{\displaystyle x=x_{c}+x'\ \ or\ \ x_{c}=x-x'}
2470:https://ieeexplore.ieee.org/document/5382047/
2337:Merlin, P. M.; Farber, D. J. (January 1975).
2320:Image Shape Extraction using Interval Methods
311:
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1153:. Then the improved Accumulator is given by
799:i.e., all the indices are incremented by –
713:and increment all the corresponding points
2411:section 4.3.4, Sonka et al., section 5.2.6
2081:, compute the candidate reference points:
1874:, compute the candidate reference points:
927:, respectively, then for a scaling factor
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2193:, then the object contour is located at
1924:, then the object contour is located at
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877:and the reference points for the shapes
807:are found, and then they are rotated by
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1649:Combining the above equations we have:
737:Generalization of scale and orientation
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2054:(1) Initialize the Accumulator table:
1843:(1) Initialize the Accumulator table:
1756:{\displaystyle y_{c}=y+r\sin(\alpha )}
1699:{\displaystyle x_{c}=x+r\cos(\alpha )}
827:is added to each vector in the table.
784:and this transformation is denoted by
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780:. Also, if the object is rotated by
410:Theory of generalized Hough transform
1859:, retrieve from the R-table all the
776:i.e., all the vectors are scaled by
342:in 1981, is the modification of the
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803:modulo 2π, the appropriate vectors
24:
1897:(2.3) Increase counters (voting):
1782:(1) Pick a reference point (e.g.,
935:, the generalized Hough transform
831:Alternate way using pairs of edges
215:Affine invariant feature detection
25:
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2317:Jaulin, L.; Bazeille, S. (2013).
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153:Maximally stable extremal regions
110:Hessian feature strength measures
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709:in the image, find the gradient
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2066:(2.1) Using its gradient angle
1855:(2.1) Using the gradient angle
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2343:IEEE Transactions on Computers
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2396:J.A. Heather, Xue Dong Yang,
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2287:Outline of object recognition
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487:as shown in the image. Store
148:Determinant of Hessian (DoH)
143:Difference of Gaussians (DoG)
2218:Advantages and disadvantages
495:. Notice that each index of
207:Generalized structure tensor
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332:generalized Hough transform
186:Generalized Hough transform
138:Laplacian of Gaussian (LoG)
18:Generalized Hough transform
10:
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2272:Randomized Hough transform
386:
2209:, and has been scaled by
717:in the accumulator array
467:Choose a reference point
2234:It is tolerant to noise.
2060:(2) For each edge point
1849:(2) For each edge point
1767:Constructing the R-table
1064:is given as a composite
499:may have many values of
430:is its orientation, and
395:and did not account for
381:relaxed set of equations
2355:10.1109/t-c.1975.224087
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346:using the principle of
223:Affine shape adaptation
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287:Implementation details
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1178:recursive subdivision
1172:Spatial decomposition
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105:Level curve curvature
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705:For each edge pixel
463:Building the R-table
2409:Ballard and Brown,
2070:, retrieve all the
1891:= y + r sin(α)
1882:= x + r cos(α)
1838:Canny edge detector
1778:Canny edge detector
701:Object localization
241:Feature description
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31:Feature detection
16:(Redirected from
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2100:= r sin(α)
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442:describes two
437:
433:
411:
408:
388:
385:
326:
325:
323:
322:
315:
308:
300:
297:
296:
295:
294:
289:
284:
276:
275:
269:
268:
267:
266:
261:
256:
251:
243:
242:
238:
237:
236:
235:
233:Hessian affine
230:
225:
217:
216:
212:
211:
210:
209:
204:
196:
195:
191:
190:
189:
188:
183:
175:
174:
170:
169:
163:
162:
161:
160:
155:
150:
145:
140:
132:
131:
129:Blob detection
125:
124:
123:
122:
117:
112:
107:
102:
100:Shi and Tomasi
97:
89:
88:
82:
81:
80:
79:
74:
69:
64:
59:
54:
49:
41:
40:
38:Edge detection
34:
33:
9:
6:
4:
3:
2:
2506:
2495:
2492:
2491:
2489:
2480:
2476:
2473:
2471:
2467:
2465:
2461:
2459:
2455:
2453:
2449:
2446:
2445:
2433:
2427:
2418:
2412:
2406:
2399:
2393:
2386:
2380:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2333:
2322:
2321:
2313:
2307:
2302:
2298:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2264:
2258:
2246:
2245:
2242:Disadvantages
2236:
2233:
2230:
2227:
2226:
2212:
2208:
2204:
2192:
2188:
2186:
2182:
2170:
2167:
2165:
2150:
2148:
2133:
2132:
2130:
2118:
2117:
2115:
2103:
2101:
2094:
2092:
2085:
2084:
2083:
2082:
2080:
2076:
2073:
2069:
2065:
2064:
2063:
2059:
2057:
2053:
2052:
2049:
2030:
2028:
2009:
2007:
1996:
1994:
1983:
1981:
1970:
1968:
1957:
1956:
1955:
1953:
1949:
1944:
1943:
1942:General case:
1935:
1923:
1919:
1918:
1916:
1912:
1904:
1901:
1900:
1899:
1898:
1896:
1892:
1885:
1883:
1876:
1875:
1873:
1869:
1866:
1862:
1858:
1854:
1853:
1852:
1848:
1846:
1842:
1839:
1835:
1834:
1833:
1832:
1825:
1821:
1817:
1805:
1803:
1799:
1796:
1793:
1781:
1779:
1775:
1771:
1770:
1769:
1768:
1747:
1741:
1738:
1735:
1732:
1729:
1726:
1721:
1717:
1709:
1708:
1690:
1684:
1681:
1678:
1675:
1672:
1669:
1664:
1660:
1652:
1651:
1650:
1630:
1624:
1621:
1618:
1615:
1609:
1606:
1603:
1597:
1594:
1591:
1588:
1584:
1581:
1560:
1556:
1551:
1548:
1544:
1538:
1535:
1532:
1526:
1523:
1516:
1515:
1497:
1491:
1488:
1485:
1482:
1479:
1473:
1470:
1467:
1461:
1458:
1455:
1452:
1448:
1445:
1424:
1420:
1415:
1412:
1408:
1402:
1399:
1396:
1390:
1387:
1380:
1379:
1363:
1360:
1356:
1353:
1350:
1345:
1341:
1331:
1328:
1318:
1315:
1311:
1306:
1302:
1298:
1295:
1288:
1287:
1271:
1268:
1264:
1261:
1258:
1253:
1249:
1239:
1236:
1226:
1223:
1219:
1214:
1210:
1206:
1203:
1196:
1195:
1194:
1186:
1183:
1179:
1169:
1167:
1160:
1135:
1131:
1127:
1124:
1116:
1112:
1106:
1101:
1098:
1095:
1091:
1087:
1081:
1075:
1067:
1063:
1046:
1041:
1034:
1024:
1020:
1015:
1009:
1004:
1001:
998:
994:
989:
983:
979:
974:
968:
964:
960:
955:
951:
942:
934:
930:
926:
919:
912:
905:
901:
894:
887:
880:
876:
869:
862:
849:
846:
842:
838:
837:a = (y, s, θ)
828:
826:
822:
818:
814:
810:
806:
802:
798:
790:
783:
779:
775:
767:
760:
756:
752:
748:
744:
734:
732:
728:
724:
720:
716:
712:
708:
693:
690:
687:
686:
683:
656:
654:
651:
648:
647:
644:
617:
615:
612:
609:
608:
605:
578:
575:
572:
571:
568:
559:
557:
551:
548:
547:
540:
536:
534:
522:
502:
498:
494:
490:
486:
482:
478:
474:
470:
460:
457:
453:
449:
445:
441:
429:
425:
421:
417:
407:
404:
402:
398:
394:
384:
382:
377:
373:
368:
364:
361:
357:
353:
349:
345:
341:
337:
333:
321:
316:
314:
309:
307:
302:
301:
299:
298:
293:
290:
288:
285:
283:
280:
279:
278:
277:
274:
271:
270:
265:
262:
260:
257:
255:
252:
250:
247:
246:
245:
244:
240:
239:
234:
231:
229:
228:Harris affine
226:
224:
221:
220:
219:
218:
214:
213:
208:
205:
203:
200:
199:
198:
197:
193:
192:
187:
184:
182:
179:
178:
177:
176:
172:
171:
168:
165:
164:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
135:
134:
133:
130:
127:
126:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
92:
91:
90:
87:
84:
83:
78:
77:Roberts cross
75:
73:
70:
68:
65:
63:
60:
58:
55:
53:
50:
48:
45:
44:
43:
42:
39:
36:
35:
32:
29:
28:
19:
2426:
2417:
2405:
2392:
2379:
2349:(1): 96–98.
2346:
2342:
2332:
2319:
2312:
2301:
2255:
2252:Related work
2210:
2206:
2194:
2190:
2184:
2168:
2151:
2134:
2120:
2105:
2095:
2086:
2078:
2071:
2067:
2061:
2055:
2031:
2010:
1997:
1984:
1971:
1967:) → (x″, y″)
1958:
1951:
1947:
1945:
1941:
1940:
1925:
1921:
1914:
1902:
1886:
1877:
1871:
1864:
1860:
1856:
1850:
1844:
1830:
1829:
1823:
1819:
1807:
1801:
1800:(3) Compute
1783:
1766:
1765:
1648:
1192:
1175:
1162:
1154:
1061:
943:is given by
936:
932:
928:
921:
914:
907:
903:
896:
889:
882:
878:
871:
864:
857:
855:
844:
840:
836:
834:
824:
820:
816:
812:
808:
804:
800:
792:
785:
781:
777:
769:
762:
758:
754:
750:
746:
742:
740:
730:
726:
722:
718:
714:
710:
706:
704:
657:
652:
618:
613:
579:
560:
552:
524:
504:
500:
496:
492:
488:
484:
476:
472:
468:
466:
455:
451:
447:
431:
427:
419:
415:
413:
405:
390:
372:binary image
369:
365:
335:
331:
329:
57:Differential
2002:by x″ and y
1998:Replacing x
1066:convolution
475:, compute
393:translation
273:Scale space
2383:L. Davis,
2293:References
2223:Advantages
2160:sin(Θ) + y
2143:cos(Θ) – y
2044:sin(Θ) + y
2023:cos(Θ) – y
1989:sin(Θ) + y
1976:cos(Θ) – y
1831:Detection:
815:such that
797:= Rot{R,θ}
745:and scale
444:orthogonal
2363:0018-9340
1748:α
1742:
1691:α
1685:
1631:α
1625:
1610:α
1607:−
1604:π
1598:
1539:α
1536:−
1533:π
1527:
1498:α
1492:
1483:−
1474:α
1471:−
1468:π
1462:
1403:α
1400:−
1397:π
1391:
1357:−
1265:−
1128:−
1092:∑
1035:ϕ
995:⋃
984:θ
956:ϕ
751:(y, s, θ)
485:r = y – x
416:a={y,s,θ}
403:changes.
2488:Category
2371:27723442
2261:See also
2191:A > T
2164:cos(Θ))s
2156:= y - (x
2147:sin(Θ))s
2139:= x - (x
2048:cos(Θ))s
2040:= y - (x
2027:sin(Θ))s
2019:= x - (x
1993:cos(Θ))s
1980:sin(Θ))s
1922:A > T
1585:′
1552:′
1449:′
1416:′
1364:′
1319:′
1272:′
1227:′
1182:quadtree
725:, i.e.,
481:gradient
397:rotation
292:Pyramids
72:Robinson
2125:; s ≤ s
2110:; Θ ≤ Θ
2006:by y″:
1985:y″ = (x
1972:x″ = (x
823:, i.e.
821:R(ɸ)+ z
817:y-ỹ = z
791:, then
774:= sR(ɸ)
768:. then
674:)... (r
635:)... (r
596:)... (r
387:History
360:ellipse
67:Prewitt
52:Deriche
2475:MATLAB
2448:OpenCV
2369:
2361:
2079:(α, r)
2072:(α, r)
2062:(x, y)
1903:++A(])
1861:(α, r)
1851:(x, y)
1826:table.
1578:
1575:
1567:
1564:
1442:
1439:
1431:
1428:
1338:
1335:
1326:
1323:
1246:
1243:
1234:
1231:
479:, the
432:s = (s
424:origin
418:where
356:circle
2367:S2CID
2324:(PDF)
2169:++(A)
2129:; s++
2121:s = s
2114:; Θ++
2106:Θ = Θ
1872:(α,r)
1159:= A*H
920:, ..
895:, ..
870:, ..
513:), (y
454:and
401:scale
376:locus
115:SUSAN
62:Sobel
47:Canny
2359:ISSN
2347:C-24
2119:for(
2104:for(
1824:R(ɸ)
1062:H(y)
902:are
759:R(ɸ)
727:r(ɸ)
694:...
666:) (r
627:) (r
588:) (r
477:ɸ(x)
399:and
352:line
330:The
259:GLOH
254:SURF
249:SIFT
158:PCBR
120:FAST
2351:doi
2199:, y
2189:If
2127:max
2123:min
2112:max
2108:min
1963:, y
1930:, y
1920:If
1822:in
1812:, y
1788:, y
1739:sin
1682:cos
1622:sin
1595:sin
1524:sin
1489:cos
1459:cos
1388:cos
941:(ɸ)
757:by
715:x+r
691:...
688:...
678:, α
670:, α
662:, α
653:2Δɸ
639:, α
631:, α
623:, α
600:, α
592:, α
584:, α
529:, α
517:– y
509:– x
505:((x
436:, s
336:GHT
264:HOG
2490::
2365:.
2357:.
2345:.
2341:.
2195:(x
2131:)
2116:)
1959:(x
1954::
1926:(x
1917:.
1808:(x
1784:(x
1571:or
1435:or
913:,
906:,
888:,
881:,
863:,
680:3k
676:3k
672:32
668:32
664:31
660:31
658:(r
641:2m
637:2m
633:22
629:22
625:21
621:21
619:(r
614:Δɸ
602:1n
598:1n
594:12
590:12
586:11
582:11
580:(r
531:ij
527:ij
525:(r
521:))
519:ij
511:ij
358:,
354:,
2373:.
2353::
2213:.
2211:s
2207:Θ
2203:)
2201:c
2197:c
2185:A
2162:′
2158:′
2154:c
2152:y
2145:′
2141:′
2137:c
2135:x
2098:′
2096:y
2089:′
2087:x
2068:ɸ
2056:A
2046:′
2042:′
2038:c
2034:c
2032:y
2025:′
2021:′
2017:c
2013:c
2011:x
2004:′
2000:′
1991:′
1987:′
1978:′
1974:′
1965:′
1961:′
1952:s
1948:Θ
1934:)
1932:c
1928:c
1915:A
1889:c
1887:y
1880:c
1878:x
1867:.
1865:ɸ
1857:ɸ
1845:A
1840:.
1820:ɸ
1816:)
1814:c
1810:c
1802:ɸ
1794:)
1792:)
1790:c
1786:c
1751:)
1745:(
1736:r
1733:+
1730:y
1727:=
1722:c
1718:y
1694:)
1688:(
1679:r
1676:+
1673:x
1670:=
1665:c
1661:x
1634:)
1628:(
1619:r
1616:=
1613:)
1601:(
1592:r
1589:=
1582:y
1561:r
1557:/
1549:y
1545:=
1542:)
1530:(
1501:)
1495:(
1486:r
1480:=
1477:)
1465:(
1456:r
1453:=
1446:x
1425:r
1421:/
1413:x
1409:=
1406:)
1394:(
1361:y
1354:y
1351:=
1346:c
1342:y
1332:r
1329:o
1316:y
1312:+
1307:c
1303:y
1299:=
1296:y
1269:x
1262:x
1259:=
1254:c
1250:x
1240:r
1237:o
1224:x
1220:+
1215:c
1211:x
1207:=
1204:x
1165:s
1163:A
1157:s
1155:A
1141:)
1136:i
1132:y
1125:y
1122:(
1117:i
1113:h
1107:N
1102:1
1099:=
1096:i
1088:=
1085:)
1082:y
1079:(
1076:H
1047:}
1042:]
1038:)
1032:(
1025:k
1021:s
1016:R
1010:N
1005:1
1002:=
999:k
990:[
980:T
975:{
969:s
965:T
961:=
952:R
939:s
937:R
933:θ
929:s
924:n
922:y
917:2
915:y
910:1
908:y
904:y
899:N
897:S
892:2
890:S
885:1
883:S
879:S
874:N
872:S
867:2
865:S
860:1
858:S
845:a
841:θ
825:z
813:ỹ
809:θ
805:r
801:θ
795:θ
793:T
788:θ
786:T
782:θ
778:s
772:s
770:T
765:s
763:T
755:S
747:s
743:θ
731:A
723:ɸ
719:A
711:ɸ
707:x
682:)
649:3
643:)
610:2
604:)
576:0
573:1
565:i
563:ɸ
561:R
555:i
553:ɸ
549:i
533:)
515:c
507:c
501:r
497:ɸ
493:ɸ
489:r
473:x
469:y
456:θ
452:s
448:y
440:)
438:y
434:x
428:θ
420:y
334:(
319:e
312:t
305:v
20:)
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