Patent Application: US-66081903-A

Abstract:
a system and process for determining the similarity in the shape of objects is presented that generates a novel shape representation called a directional histogram model . this shape representative captures the shape variations of an object with viewing direction , using thickness histograms . the resulting directional histogram model is substantially invariant to scaling and translation . a matrix descriptor can also be derived by applying the spherical harmonic transform to the directional histogram model . the resulting matrix descriptor is substantially invariant to not only scaling and translation , but rotation as well . the matrix descriptor is also robust with respect to local modification or noise , and able to readily distinguish objects with different global shapes . the typical applications of the directional histogram model and matrix descriptor include recognizing 3d solid shapes , measuring the similarity between different objects and shape similarity based object retrieval .

Description:
in the following description of the preferred embodiments of the present invention , reference is made to the accompanying drawings which form a part hereof , and in which is shown by way of illustration specific embodiments in which the invention may be practiced . it is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention . before providing a description of the preferred embodiments of the present invention , a brief , general description of a suitable computing environment in which the invention may be implemented will be described . fig1 illustrates an example of a suitable computing system environment 100 . the computing system environment 100 is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention . neither should the computing environment 100 be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary operating environment 100 . the invention is operational with numerous other general purpose or special purpose computing system environments or configurations . examples of well known computing systems , environments , and / or configurations that may be suitable for use with the invention include , but are not limited to , personal computers , server computers , handheld or laptop devices , multiprocessor systems , microprocessor based systems , set top boxes , programmable consumer electronics , network pcs , minicomputers , mainframe computers , distributed computing environments that include any of the above systems or devices , and the like . the invention may be described in the general context of computer executable instructions , such as program modules , being executed by a computer . generally , program modules include routines , programs , objects , components , data structures , etc . that perform particular tasks or implement particular abstract data types . the invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network . in a distributed computing environment , program modules may be located in both local and remote computer storage media including memory storage devices . with reference to fig1 , an exemplary system for implementing the invention includes a general purpose computing device in the form of a computer 110 . components of computer 110 may include , but are not limited to , a processing unit 120 , a system memory 130 , and a system bus 121 that couples various system components including the system memory to the processing unit 120 . the system bus 121 may be any of several types of bus structures including a memory bus or memory controller , a peripheral bus , and a local bus using any of a variety of bus architectures . by way of example , and not limitation , such architectures include industry standard architecture ( isa ) bus , micro channel architecture ( mca ) bus , enhanced isa ( eisa ) bus , video electronics standards association ( vesa ) local bus , and peripheral component interconnect ( pci ) bus also known as mezzanine bus . computer 110 typically includes a variety of computer readable media . computer readable media can be any available media that can be accessed by computer 110 and includes both volatile and nonvolatile media , removable and nonremovable media . by way of example , and not limitation , computer readable media may comprise computer storage media and communication media . computer storage media includes both volatile and nonvolatile , removable and nonremovable media implemented in any method or technology for storage of information such as computer readable instructions , data structures , program modules or other data . computer storage media includes , but is not limited to , ram , rom , eeprom , flash memory or other memory technology , cdrom , digital versatile disks ( dvd ) or other optical disk storage , magnetic cassettes , magnetic tape , magnetic disk storage or other magnetic storage devices , or any other medium which can be used to store the desired information and which can be accessed by computer 110 . communication media typically embodies computer readable instructions , data structures , program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media . the term “ modulated data signal ” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal . by way of example , and not limitation , communication media includes wired media such as a wired network or direct wired connection , and wireless media such as acoustic , rf , infrared and other wireless media . combinations of the any of the above should also be included within the scope of computer readable media . the system memory 130 includes computer storage media in the form of volatile and / or nonvolatile memory such as read only memory ( rom ) 131 and random access memory ( ram ) 132 . a basic input / output system 133 ( bios ), containing the basic routines that help to transfer information between elements within computer 110 , such as during startup , is typically stored in rom 131 . ram 132 typically contains data and / or program modules that are immediately accessible to and / or presently being operated on by processing unit 120 . by way of example , and not limitation , fig1 illustrates operating system 134 , application programs 135 , other program modules 136 , and program data 137 . the computer 110 may also include other removable / nonremovable , volatile / nonvolatile computer storage media . by way of example only , fig1 illustrates a hard disk drive 141 that reads from or writes to nonremovable , nonvolatile magnetic media , a magnetic disk drive 151 that reads from or writes to a removable , nonvolatile magnetic disk 152 , and an optical disk drive 155 that reads from or writes to a removable , nonvolatile optical disk 156 such as a cd rom or other optical media . other removable / nonremovable , volatile / nonvolatile computer storage media that can be used in the exemplary operating environment include , but are not limited to , magnetic tape cassettes , flash memory cards , digital versatile disks , digital video tape , solid state ram , solid state rom , and the like . the hard disk drive 141 is typically connected to the system bus 121 through a nonremovable memory interface such as interface 140 , and magnetic disk drive 151 and optical disk drive 155 are typically connected to the system bus 121 by a removable memory interface , such as interface 150 . the drives and their associated computer storage media discussed above and illustrated in fig1 , provide storage of computer readable instructions , data structures , program modules and other data for the computer 110 . in fig1 , for example , hard disk drive 141 is illustrated as storing operating system 144 , application programs 145 , other program modules 146 , and program data 147 . note that these components can either be the same as or different from operating system 134 , application programs 135 , other program modules 136 , and program data 137 . operating system 144 , application programs 145 , other program modules 146 , and program data 147 are given different numbers here to illustrate that , at a minimum , they are different copies . a user may enter commands and information into the computer 110 through input devices such as a keyboard 162 and pointing device 161 , commonly referred to as a mouse , trackball or touch pad . other input devices ( not shown ) may include a microphone , joystick , game pad , satellite dish , scanner , or the like . these and other input devices are often connected to the processing unit 120 through a user input interface 160 that is coupled to the system bus 121 , but may be connected by other interface and bus structures , such as a parallel port , game port or a universal serial bus ( usb ). a monitor 191 or other type of display device is also connected to the system bus 121 via an interface , such as a video interface 190 . in addition to the monitor , computers may also include other peripheral output devices such as speakers 197 and printer 196 , which may be connected through an output peripheral interface 195 . of particular significance to the present invention , a camera 163 ( such as a digital / electronic still or video camera , or film / photographic scanner ) capable of capturing a sequence of images 164 can also be included as an input device to the personal computer 110 . further , while just one camera is depicted , multiple cameras could be included as input devices to the personal computer 110 . the images 164 from the one or more cameras are input into the computer 110 via an appropriate camera interface 165 . this interface 165 is connected to the system bus 121 , thereby allowing the images to be routed to and stored in the ram 132 , or one of the other data storage devices associated with the computer 110 . however , it is noted that image data can be input into the computer 110 from any of the aforementioned computer readable media as well , without requiring the use of the camera 163 . the computer 110 may operate in a networked environment using logical connections to one or more remote computers , such as a remote computer 180 . the remote computer 180 may be a personal computer , a server , a router , a network pc , a peer device or other common network node , and typically includes many or all of the elements described above relative to the computer 110 , although only a memory storage device 181 has been illustrated in fig1 . the logical connections depicted in fig1 include a local area network ( lan ) 171 and a wide area network ( wan ) 173 , but may also include other networks . such networking environments are commonplace in offices , enterprise wide computer networks , intranets and the internet . when used in a lan networking environment , the computer 110 is connected to the lan 171 through a network interface or adapter 170 . when used in a wan networking environment , the computer 110 typically includes a modem 172 or other means for establishing communications over the wan 173 , such as the internet . the modem 172 , which may be internal or external , may be connected to the system bus 121 via the user input interface 160 , or other appropriate mechanism . in a networked environment , program modules depicted relative to the computer 110 , or portions thereof , may be stored in the remote memory storage device . by way of example , and not limitation , fig1 illustrates remote application programs 185 as residing on memory device 181 . it will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used . the exemplary operating environment having now been discussed , the remaining part of this description section will be devoted to a description of the program modules embodying the invention . generally , the system and process according to the present invention first involves generating a directional histogram model to study the shape similarity problem of 3d objects . this novel representation is based on the depth variations with viewing direction . more particularly , in each viewing direction , a histogram of object thickness values is built to create a directional histogram model . this model is substantially invariant to translation , scaling and origin - symmetric transform , since the histograms are obtained in such a way that they are independent of the location of the object and the scale of the object . in order to also make the 3d object representation substantially orientation invariant as well , a new shape descriptor having a matrix form , called matrix descriptor , is computed from the directional histogram model by computing the spherical harmonic transform of the model . the foregoing is generally accomplished by generating a representation of an object as follows . first , for a prescribed number of directions , the thickness of the object for each of a prescribed number of parallel rays directed through the object along the direction under consideration is determined . thus , referring to fig2 a and 2b , a previously unselected one of the prescribed directions is selected ( process action 200 ). then , a previously unselected one of the parallel rays associated with the selected direction is selected ( process action 202 ). it is next determined if the selected ray transects the object being modeled ( process action 204 ). if not , process actions 202 and 204 are repeated to assess the status of another ray . if , however , the selected ray does transect the object , then in process action 206 , the transecting distance ( i . e ., the thickness ) is computed . it is next determined in process action 208 if there are any remaining rays that have not yet been considered . if there are such unconsidered rays , then process actions 202 through 208 are repeated . once all the rays have been considered , the resulting distance or thickness values are normalized such that the maximum thickness is one ( process action 210 ), and can then be uniformly quantized ( optional process action 212 ). a thickness histogram is then generated in process action 214 from the normalized values through the process of banning . next , this thickness histogram can be rescaled ( i . e ., normalized ) such that the sum of squares of its bins is one ( optional process action 216 ). once the thickness histogram has been rescaled , it is determined if all the prescribed directions have been considered ( process action 218 ). if not , then process actions 200 through 218 are repeated to generate additional thickness histograms . when all the directions have been considered , the thickness histograms associated with the prescribed directions are collectively designated as the aforementioned directional histogram model of the object ( process action 220 ). the directional histogram model could be used as the representation of the object as it is substantially invariant to translation and scaling . however , the model could be further characterized as a number of spherical functions defined on a unit sphere ( optional process action 222 ), which are subjected to a spherical harmonic transform to produce the aforementioned matrix descriptor ( optional process action 224 ). using this descriptor to represent an object has the added advantage of being substantially invariant to rotation ( i . e ., orientation ), in addition to translation and scaling . it is noted that the optional nature of the last two actions , as well as actions 212 and 216 , is indicated by the dashed line boxes in fig2 a and 2 b . the foregoing object representation can be used in a variety of applications requiring the measurement of the similarity between 3d objects . for example , these applications include 3d shape recognition , shape similarity based object retrieval systems , and 3d search engines . often the applications involve finding objects in a database that are similar to a sample object . in the context of the present 3d object representation technique , this can be accomplished as follows . referring to fig3 , a database is first created where objects of interest are characterized as the aforementioned matrix descriptors ( process action 300 ). the geometric model of a sample object is then input ( process action 302 ) and a matrix descriptor representation of the object is generated ( process action 304 ) as described above . the matrix descriptor of the sample object is compared with each of the matrix descriptors in the database , and a distance measurement for each comparison is computed ( process action 306 ). this distance measurement is indicative of the degree of similarity there is between the matrix descriptors of the compared pair and will be described in more detail later . next , a prescribed number of the objects characterized in the database that are associated with the lowest distance measurements are identified ( process action 308 ). the identified objects are then designated as being similar to the sample object ( process action 310 ). alternately , in lieu of performing actions 308 and 310 , the following actions can be performed . namely , those objects characterized in the database whose associated difference measurement falls below a difference threshold are identified ( alternate process action 312 ). these objects are then designated as being similar to the sample object ( alternate process action 314 ). the alternate nature of the last two actions is indicated in fig3 by the dotted line boxes . in some applications there is no database and it is simply desired to access the similarity of a pair of 3d objects . this can be accomplished as outlined in fig4 . first , a geometric model of each of the 3d objects being compared is input ( process action 400 ). a matrix descriptor representation of each object is then generated ( process action 402 ) in the manner described previously . the matrix descriptors of the objects are compared to each other , and a distance measurement is computed between them ( process action 404 ). as before , this distance measurement is indicative of the degree of similarity there is between the matrix descriptors of the compared pair . objects whose associated difference measurement falls below a difference threshold are designated as being similar to each other ( process action 406 ). the following sections will now describe the foregoing 3d object representation technique in more detail . this section will describe the directional histogram model for 3d shapes . the goal of the directional histogram model is to develop an invariant and expressive representation suited for 3d shape similarity estimation . to construct the directional histogram model , a distribution of sampling directions is chosen . for each sampling direction , a histogram of object extent or thickness is computed using parallel rays . for each ray , the thickness is defined as the distance between the nearest and farthest points of intersection with the object surface . the directional histogram model can be represented by a 3 - d function h ( θ , φ , μ ): [ 0 , π ]×[ 0 , 2π ]×[ 0 , 1 ] r , where θ , φ are the angular parameters for direction . for each ( θ , φ ), the direction vector is ( cos φ sin θ , sin φ sin θ , cos θ ), and h θ , φ ( μ )≡ h ( θ , φ , μ ) is the thickness distribution of the object viewed from the direction ( θ , 100 ). note that each thickness histogram is also normalized with respect to the thickest value to ensure scale invariance . in tested embodiments of the present inventions , the sampling directions were computed as , { ( θ i , ϕ j ) | θ i = ( i + 0 . 5 ) ⁢ π n s , ϕ j = ( j + 0 . 5 ) ⁢ 2 ⁢ ⁢ π n s ) , 0 ≤ i , j & lt ; n s } , ( 1 ) where n s ( an integer greater than zero ) is the sampling rate . since two opposing sampling directions will produce the same thickness values , the directional histogram model is symmetric about the origin , i . e ., more particularly , one way of generating a directional histogram model is as follows . referring to fig5 a and 5b , a previously unselected sampling direction ( θ , φ ) is selected ( process action 500 ). a prescribed number of rays in the selected sampling direction are then generated ( process action 502 ). a ray is a line in space of infinite length . for a given direction computed using eq . ( 1 ), the sampling rays are shifted versions of each other . they pass through a 2d n w by n w sampling grid or window , with the 2d sampling window being perpendicular to the direction of the sampling rays . the sampling window covers the visual extent of the object being modeled . a side view of the rays is shown in fig7 a . while any number of rays can be employed , in tested embodiments , the number of rays was tied to the size of n w , as measured in pixels , such that there was no more than one ray for each pixel . as an example , it is noted that n w ranged from 16 to 512 in various experiments with the tested embodiments . thus , the number of rays could range from 256 to 262 , 144 . referring once again to fig5 a , a previously unselected one of the generated rays is then selected ( process action 504 ) and it is determined if the selected ray transects the object being modeled ( process action 506 ). if the ray does not transect the object , then another ray is selected and processed by repeating actions 504 and 506 . otherwise , in process action 508 , the 3d coordinates of the nearest point of intersection p n of the selected ray with the object in the selected direction is identified . in addition , in process action 510 , the 3d coordinates of the furthest point of intersection p f of the selected ray with the object in the selected direction is identified . the distance μ between p n and p f is then computed ( process action 512 ). this distance represents the thickness of the object as transected by the selected ray . next , in process action 514 , it is determined if there are any remaining previously unselected rays generated for the selected direction . if so , then actions 504 through 514 are repeated . however , if all the rays have been processed , then the distance values μ associated with the selected direction are normalized ( process action 516 ). this normalizing action is performed because the distance values are scale - dependent . in order to obtain a scale - independent representation , the distance values are normalized . this can be accomplished by dividing each of the values by the maximum distance value μ max ( θ , φ ). a thickness histogram h θ , φ ({ circumflex over ( μ )}) for the selected direction is then constructed using the normalized distance values { circumflex over ( μ )} ( process action 518 ). it is next determined if there are any remaining previously unselected sampling directions ( process action 520 ). if there are , process actions 500 through 520 are repeated for the each of the remaining directions to produce additional thickness histograms associated with the object being modeled . if , however , all the directions have been considered , then in process action 522 , the thickness histograms computed for each of the selected directions are designated as the directional histogram model for the object , and the process ends . the computation of the thickness histogram can also be accelerated using commercially - available graphics hardware ( e . g ., graphics accelerator cards ). for a given sampling direction , the front of the object is rendered and its depth values are read . this is then repeated for the back of the object . the thickness is the difference between the front and back depth values . more particularly , referring to fig6 a and 6b , one way of accomplishing the accelerated approach is as follows . first , a previously unselected sampling direction ( θ , φ ) is selected ( process action 600 ). the orthogonal projection of the object being modeled is then set in the selected direction ( process action 602 ), and the front part of the object associated with the selected direction is rendered ( process action 604 ). the depth value b f corresponding to each pixel of the object &# 39 ; s front part is then read ( process action 606 ). the back part of the object associated with the selected direction is then rendered ( process action 608 ) and the depth value b b corresponding to each pixel are read ( process action 610 ). a previously unselected pair of corresponding pixels of the front and back parts is selected next ( process action 612 ), and the distance between the depth values associated therewith is computed as b b − b f ( process action 614 ). it is then determined if all the corresponding pixel pairs have been considered ( process action 616 ). if not , then process actions 612 and 616 are repeated until a distance value has been computed for each pixel pair . when all the pixel pairs have been considered , then the distance values μ associated with the selected direction are normalized ( process action 618 ). as before , this can be accomplished by dividing each of the values by the maximum distance value μ max ( θ , φ ). a thickness histogram h θ , φ ({ circumflex over ( μ )}) for the selected direction is then constructed using the normalized distance values { circumflex over ( μ )} ( process action 620 ). it is next determined if there are any remaining previously unselected sampling directions ( process action 622 ). if there are , process actions 600 through 622 are repeated for the each of the remaining directions to produce additional thickness histograms associated with the object being modeled . if , however , all the directions have been considered , then in process action 624 , the thickness histograms computed for each of the selected directions are collectively designated as the directional histogram model for the object , and the process ends . it is noted that in the accelerated process , the thickness of the object in each direction is defined on a pixel basis with a different distance value being computed for each pair of corresponding pixel locations in the front and back rendering of the object . thus , in essence each corresponding pixel pair defines a ray in the selected direction similar to the previously described process . it is further noted that in the foregoing accelerated procedure , the object &# 39 ; s back part can be rendered by setting the initial z - buffer values as 1 and the depth comparison function to let the greater z - values pass , e . g ., gldepthfunc ( gl_gequal ) in an opengl implementation . to simplify future similarity computations between objects represented with a directional histogram model , the distance values μ can be uniformly quantized into m integer values after being normalized . m = 64 worked well in the tested embodiments . further , to facilitate the directional histogram model &# 39 ; s conversion into the aforementioned matrix descriptor , each thickness histogram making up the model can also be normalized such that ∫  h θ , ϕ ⁡ ( μ )  2 ⁢ ⅆ μ = 1 , i . e . , ∑ k = 0 m - 1 ⁢  h θ , ϕ ⁡ ( k m )  2 = 1 . ( 2 ) invariance properties 4 and 5 , which will be discussed shortly , both rely on this histogram normalization . fig7 ( a ) and ( b ) illustrate the sampling geometry employed to calculate a histogram in direction ( θ , φ ). in fig7 ( a ), for a particular direction ( θ , φ ), a group of parallel rays is shown which traverse a shape ( i . e ., a rabbit ). the distance on each ray from the first intersection point to the last intersection point is shown as solid lines , whereas outside the shape the rays are shown as dashed lines . it is noted that not all the sampling rays are shown for the sake of clarity . fig7 ( b ) is an example histogram representing a possible thickness distribution for the sampling of the shape at the direction ( θ , φ ) under consideration . based on the model construction process , it is clear that the directional histogram model is invariant to translation and scaling . but it is still orientation dependent . to remove the dependence on orientation , a new representation will be derived from the directional histogram model , called a matrix descriptor . to create this matrix descriptor , the directional histogram model is characterized as m number of spherical functions h k ( θ , φ ) defined on a unit sphere : h k ⁡ ( θ , ϕ ) ≡ h ⁡ ( θ , ϕ , k m ) . h k ⁡ ( θ , ϕ ) = ∑ l = 0 ∞ ⁢ ∑ m = - l l ⁢ h klm ⁢ y l ⁢ ⁢ m ⁡ ( θ , ϕ ) , ( 3 ) where h klm =& lt ; h k ( θ , φ ), y lm ( θ , φ )& gt ;, and y lm ( θ , φ ) are the spherical harmonic functions . one of the most useful properties of y lm ( θ , φ ) is that y lm ( θ + α , φ + β )= σ mm ′=− l l d mm ′ l ( α ) e lmβ y lm ′ ( θ , φ ), where the combination coefficients satisfy : σ m ′=− l l | d mm ′ l ( α )| 2 = 1 . since σ l = 0 l σ m =− l l h klm y lm ( θ , φ ) converges to h k ( θ , φ ) as l →∞, it can be assumed that h k ( θ , φ ) is bandwidth limited for simplicity . assuming that the bandwidth of h k ( θ , φ ) is less than n , then : based on the spherical harmonic coefficients h klm , the matrix descriptor m is defined as m =( a lk ) m × n , where a lk = ∑ m = - l l ⁢  h klm  2 . ( 5 ) in the above definition , a lk represents the energy sum of h k ( θ , φ ) at band l . therefore , the matrix descriptor gives the energy distribution of the directional histogram model over each ( discrete ) thickness value and each band index — a distribution that does not change with rotation . in this section , the matrix descriptor is analyzed in regard to its invariance properties . for the purposes of this analysis , the matrix descriptor is considered to be separate from the directional histogram model , even though the matrix descriptor is derived from the directional histogram model . the first property of the matrix descriptor to be analyzed will show that : property 1 : the matrix descriptor m of a 3d object is invariant to rotation , translation and scaling . since the directional histogram model is invariant to translation and scaling as described previously , the matrix descriptor is invariant to translation and scaling as well . thus only orientation invariance needs to be explained . suppose the object is rotated by angles ( α , β ), then the directional histogram model of the rotated object is h ′( θ , φ , μ )= h ( θ + α , φ + β , μ ). therefore , h klm ′ = 〈 h k ′ ⁡ ( θ + α , ϕ + β ) , y l ⁢ ⁢ m ⁡ ( · ) 〉 = 〈 ∑ s = 0 n ⁢ ∑ t = - s s ⁢ h kst ⁢ y st ⁡ ( θ + α , ϕ + β ) , y l ⁢ ⁢ m ⁡ ( · ) 〉 = 〈 ∑ s = 0 n ⁢ ∑ t = - s s ⁢ h kst ⁢ ∑ r = - s s ⁢ d tr s ⁡ ( α ) ⁢ ⁢ ⅇ ⅈ ⁢ ⁢ t ⁢ ⁢ β ⁢ y sr ⁡ ( · ) , y l ⁢ ⁢ m ⁡ ( · ) 〉 = ∑ s = 0 n ⁢ ∑ t = - s s ⁢ ∑ r = - s s ⁢ h kst ⁢ d tr s ⁡ ( α ) ⁢ ⁢ ⅇ ⅈ ⁢ ⁢ t ⁢ ⁢ β ⁢ 〈 y sr ⁡ ( · ) , y l ⁢ ⁢ m ⁡ ( · ) 〉 = ∑ t = - l l ⁢ h klt ⁢ d tm l ⁡ ( α ) ⁢ ⁢ ⅇ ⅈ ⁢ ⁢ t ⁢ ⁢ β , where “·” denotes θ , φ . using the orthogonality property of e itβ ,  h klm ′  2 = ⁢ ∑ t = - l l ⁢  h klt  2 ⁢  d tm l ⁡ ( α )  2 ∑ m = - l l ⁢  h klm ′  2 = ⁢ ∑ m = - l l ⁢ ∑ t = - l l ⁢  h klt  2 ⁢  d tm l ⁡ ( α )  2 = ⁢ ∑ t = - l l ⁢  h klt  2 ⁢ ⁢ ∑ m = - l l ⁢  d tm l ⁡ ( α )  2 = ⁢ ∑ t = - l l ⁢  h klt  2 = ⁢ ∑ m = - l l ⁢  h klm  2 . since a ′ lk = a lk and m = m ′, the matrix descriptor of an object is invariant to rotation . let m i be the matrix descriptor derived from a directional histogram model h i , i = 0 . 1 . if m 0 = m 1 , then h 0 and h 1 are equivalent models , denoted by property 2 : the matrix descriptor m of a 3d object is invariant to origin - symmetric transform and mirror transform . since the directional histogram model of a 3 - d object is origin - symmetric , h ( θ , φ , μ )= h (− θ + π , φ + π , μ ). then , h ( θ , φ , μ )˜ h (− θ + π , φ + π , μ ), and so the matrix descriptor m is invariant to origin - symmetric transform . to show the invariance to mirror transform , it can be assumed that the mirror is the x - y plane without loss of generality according to property 1 . let h ′( θ , φ , μ ) be the directional histogram model of the mirrored object . then property 3 : if m =( a lk ) m × n is a matrix descriptor . then , a lk = 0 , if l is odd . when l is odd , y lm (− θ + π , φ + π )=− y lm ( θ , φ ). since h k ( θ , φ )= h k (− θ + π , φ + π ), then h klm = 〈 h k ⁡ ( θ , ϕ ) , y l ⁢ ⁢ m ⁡ ( θ , ϕ ) 〉 = 〈 h k ⁡ ( - θ + π , ϕ + π ) , - y l ⁢ ⁢ m ⁡ ( - θ + π , ϕ + π ) 〉 = - h klm . therefore , h klm = 0 , and a lk =√{ square root over ( σ m =− l l | h klm | 2 )}= 0 . property 4 : the squared sum of the matrix descriptor elements is 1 . ∑ l = 0 n - 1 ⁢ ∑ k = 0 m - 1 ⁢ a lk 2 = ∑ k = 0 m - 1 ⁢ ∑ l = 0 n - 1 ⁢ ∑ m = - l l ⁢  h klm  2 = ∑ k = 0 m - 1 ⁢ ∮ s ⁢  ℋ k ⁡ ( θ , ϕ )  2 ⁢ ⅆ s = ∮ s ⁢ ∑ k = 0 m - 1 ⁢  ℋ k ⁡ ( θ , ϕ )  2 ⁢ ⅆ s = ∮ s ⁢ ∑ k = 0 m - 1 ⁢  ℋ θ , ϕ ⁡ ( k m )  2 ⁢ ⅆ s = ∮ s ⁢ ⅆ s = 1 , where s denotes the unit sphere , and ◯∫ s ds = 1 is assumed in the spherical harmonic analysis . let o 1 , o 2 be two 3d objects . the similarity between o 1 and o 2 can be measured using the norm of their matrix descriptors &# 39 ; difference m ( o 1 )−( o 2 ): d ( o 1 , o 2 )=∥ m ( o 1 )− m ( o 2 )∥. ( 6 ) any matrix norm can be used , however , for the following description it will be assumed the matrix norm employed is the l p norm , with p = 2 . let v l be the l - th row vector in a shape matrix . note that the v l represents the energy of the directional histogram model at l - th frequency , and can be weighted when calculating the object distance . when using the l p norm , the weighted form of distance function can be represented as d p ⁡ ( o 1 , o 2 ) = ( ∑ l = 0 m ⁢ ω l ⁢ ⁢  v 1 ⁢ l - v 2 ⁢ l  p ) 1 / p , ( 7 ) where v jl is the l - th row vector in the shape matrix of object o j , ω l & gt ; 0 are the weights , and ∥·∥ is the l p norm of a vector . by adjusting the weights , it is possible to emphasize the importance of objects at some frequency when evaluating the shape similarity . in tested embodiments , p = 2 was chosen and all weights ω l = 1 . with this choice , the following property on the d 2 distance function holds . property 5 : the d 2 distance between any two objects is between 0 and √{ square root over ( 2 )}. since the elements in the matrix descriptor are all positive , d 2 2 ( o 1 , o 2 )= σ l , k ( a 1lk − a 2lk ) 2 & lt ; σ l , k ( a 1lk 2 + a 2lk 2 ). then , d 2 2 ( o 1 , o 2 )& lt ; 2 , according to property 4 . therefore d 2 ( o 1 , o 2 )& lt ;√{ square root over ( 2 .)} in this section , the performance of the directional histogram model is addressed . first , the computational cost to generate the directional histogram model will be examined . assume a square window , with window size n w ( i . e ., window width ), is used to render the object in the previously described hardware accelerated approach . recall that n s is the number of angles θ and φ ( i . e ., sampling rate ). for each direction θ i , φ j , the object is rendered twice , and the depth buffer is read twice as well . fortunately , only half of all the sampling directions are needed , since opposing directions produce the same thickness histograms as shown by property 2 . as a result , the bulk of time cost is t = n s 2 ( t b ( n w )+ t r ( n w )), where t b ( n w ) is the time cost to read the depth buffer values from a n w × n w window and t r ( n w ) is the render time in the same window . usually , t r ( n w ) is roughly proportional to the object &# 39 ; s face number n f , i . e ., t r ( n w )≈ λ n w n f , where λ n w is a constant . therefore , t ≈ n s 2 ( t b ( n w )+ λ n w n f ). this is verified by the performance results shown in fig8 ( a ) and 8 ( b ). in these figures , many objects are used for testing , each of which corresponds to a curve . for curves from bottom to top , the vertex number of the corresponding object ranges uniformly from 5 , 000 to 100 , 000 . in fig8 ( a ) the execution time is plotted against the sample rate at a window size of n w = 128 . in fig8 ( b ) the execution time is plotted against the window size at a sampling rate of n s = 64 . the curves in fig8 ( a ) and 8 ( b ) show that the execution time is approximately proportional to the squared sampling rate and the window size . for a given sampling direction , the number of intersecting rays used to sample the object thickness distribution is proportional to the squared window size n w 2 . it is reasonable to expect that a higher ray density should produce more accurate thickness distribution . if the ray density is too coarse , the resulting thickness distribution and the final matrix descriptor will be strongly dependent on the location of the object as it is being sampled . the relationship between the error of the matrix descriptor introduced by perturbing the object and the window size n w was examined extensively in tested embodiments of the present invention . a 3d dragon model was first simplified to generate objects of varying complexity . fig9 ( a )-( c ) show three ( out of nine ) simplified models of the dragon . the complexity of 3 - d object was measured by its average edge length ( normalized with respect to the diameter of the object bounding sphere ). for each window size n w = 32 / 64 / 128 / 256 and for each dragon object , the matrix descriptor is computed 50 times , each time with the model randomly perturbed . these matrix descriptors were then compared with the reference matrix descriptor computed without disturbing the model , and perturbation error was analyzed with the usual statistics of mean and standard deviation . ( the matrix is compared using the l 2 norm in tested embodiments ). the results of the foregoing test are summarized in fig1 ( a ) and 10 ( b ), which show the relationship between the matrix descriptor and window size . more particularly , fig1 ( a ) plots perturbation error ( with standard deviation bars ) against the average edge length for different window sizes ( n w ), and fig1 ( b ) plots perturbation error against the ray interval ( 1 / n w ). fig1 ( a ) shows that the perturbation error is quite constant for different object complexity . fig1 ( b ) shows an interesting trend of the perturbation error being roughly proportional to the ray interval ( i . e ., inversely proportional to the window size ). while this shows that a larger window size is better , it would decrease the rendering speed and depth buffer extraction . it was found that n w = 128 is a good trade - off . for a typical object with 20k vertices in a sample database and a 128 × 128 window , it is possible to render the object and read the depth buffer at about 30 fps . therefore , the total time cost is about 1 / 30n s 2 seconds . in addition , a strict limitation on the sampling rate n s is imposed for efficiency . recall that the bandwidth of h k ( θ , φ ) is assumed as n , thus at least 2n × 2n samples are needed for h k ( θ , φ ) according to spherical harmonic analysis [ 6 ]. however , in practice , the bandwidth of h k ( θ , φ ) is not necessarily limited . using a finite number of samples would then result in loss of information ( power ). because the power distribution of h k ( θ , φ ) depends on the object shape , the number of samples needed for arbitrary objects remains unknown . this problem is analyzed as follows . first , the matrix descriptor m n is calculated for many objects at different sampling rates n s . for each object , the quality of the approximation due to the sampling rate is calculated by comparing them against m 256 ( where m 256 is used as the ground truth m ∞ , which is not possible to obtain in practice ). the results of this test shown in fig1 ( a ), where the approximation errors are plotted against the sampling rate , indicates that the approximation error drops very quickly as the sampling rate is increased . a sampling rate at least n s = 128 was determined to be a reasonable choice . however , it will take about 8 minutes to generate the directional histogram model for a typical object in the database when n s = 128 . while this is not unreasonable , it is impractical for time - critical applications such as for 3 - d search engines for web applications . as a result , the sampling rate can be reduced , albeit at the expense of fidelity of representation . in another set of tests , distances between pairs of different objects under different sampling rates ( n s = 8 / 16 / 32 / 64 / 128 / 256 ) were computed . basically , the larger the distance , the better the discrimination power . results indicate that most of the distances increase monotonically as n s is increased . to enable comparisons between different objects , the distance with respect to the distance at n s = 256 is normalized for each object pair . the graph of the average ( with standard deviation bars ) is shown in fig1 ( b ) where the normalized distance is plotted against the sampling rate . this graph clearly shows that the sampling rate n s = 16 is sufficient for shape similarity estimation , and increasing the sampling rate to more than 16 would produce only marginal improvements in accuracy at the expense of speed . note that it takes about only 8 seconds when n s = 16 . while the matrix descriptor of an object is theoretically invariant to rigid transform and scaling , it will change to some degree when the object undergoes these transformations because of the finite n s 2 directional sampling . in tested versions of the present invention , it was found that the variances introduced by scaling and translation are very small . the variances introduced by rotation are somewhat larger , but they are not significant with a sampling rate of n s ≧ 16 , as indicated by table 1 shown in fig1 , where the matrix descriptor of the rotated object is compared against with that of the original object . results under two sampling rates are listed . computational cost analysis shows that the time cost is proportional the object complexity in terms of the face / vertex number . state - of - art simplification techniques ( e . g ., [ 5 , 8 , 9 ]) are capable of simplifying large objects quickly ( typically , in only a few seconds ). using an efficient model simplification technique before sampling the thickness distributions would clearly be advantageous . generally , model simplification may introduce some error into the matrix descriptors . this is referred to as the simplification error . how much of a simplification error depends on the simplification level . to study this effect , many trials were run involving many objects , and the results are summarized in fig1 , which shows the error in the matrix descriptor introduced by simplification . the simplification is characterized by the normalized average edge length of the simplified model . the simplified object is characterized by its normalized average edge length with respect to the object bounding sphere &# 39 ; s diameter . the results show that within the range of simplification used , only small simplification errors ( e . g ., & lt ; 0 . 06 ) are obtained . note that the most simplified versions of the “ dinosaur ”, “ wolf ”, “ man ” and “ bunny ” models consist of only 1300 , 1500 , 982 , 1556 vertices respectively . in fig1 , the window size for rendering is n w = 128 . it is curious to note that the curves increase more dramatically after an average vertex distance of 0 . 0125 ( shown by a vertical line in fig1 ), which corresponds to about 1 . 5 times the ray interval . for shape similarity comparison , it was found that n s = 16 , n w = 128 is a set of good sampling parameters in terms of accuracy and efficiency , based on discussion in section 4 . 0 . using these sampling parameters , it is possible to obtain the shape comparison results indicated by table 2 shown in fig1 . in fig1 ( a )-( h ), the shape similarity between interpolated objects is compared . fig1 ( a ) represents the original object , while fig1 ( b )-( g ) represent interpolated objects between the objects of fig1 ( a ) and ( h ). the number under each object is the distance to the original object . it is interesting to note that as the object is morphed to another , the distance actually increases monotonically as expected . based on the shape similarity measurement process described in section 3 , a simple shape based 3d object retrieval prototype system can be constructed . in this prototype , a sample shape would be specified by a user , and the matrix descriptor is computed for the sample shape . the sample shape matrix descriptor is compared as described previously to a database of pre - computed and stored matrix descriptors corresponding to a number of 3d objects . in one prototype system , a prescribed number of the most similar objects , i . e ., those having the lowest distance values from the sample shape , were identified as similar objects . some input and output examples of the prototype system are shown in fig1 . in this example , a sampling rate of n s = 16 was used to speed up the comparison process . to further reduce the matrix variance due to rotation at sampling rate n s = 16 , as indicated in table 1 of fig1 , a simple pre - orientation procedure was applied . in particular , principle component analysis ( pca ) was employed to compute the major axis for automatic alignment . while the invention has been described in detail by specific reference to preferred embodiments thereof , it is understood that variations and modifications thereof may be made without departing from the true spirit and scope of the invention . for example , the above - described directional histogram model and matrix descriptor can be adapted to 2d objects as well . given a 2d object , the 2d directional histogram model is a 2d function h ( θ , φ ): [ 0 , 2π ]×[ 0 , 1 ] r , where φ is the direction parameter . for each φ , h φ ( μ )≡ h ( θ , μ ) gives the depth distribution of the object along the direction φ . applying a fourier transform to the 2d directional histogram model yields : h k ( φ )= h ( φ , k / m )= σ l = 0 ∞ h kl e − iφ . ( 8 ) the 2d object &# 39 ; s matrix descriptor can also be derived using the fourier - coefficients h kl similarly : m =( a lk ) m × n , a lk =√{ square root over (| h kl | 2 )}. in addition , the element a lk of an odd l index is zero , and the squared sum of all the matrix descriptor elements is 1 . based on the 2 - d matrix descriptor , the distance function d p is also well - defined for 2 - d object , and ranges from 0 to √{ square root over ( 2 )}. the distance function d p is also well defined for 2d objects with the use of 2d matrix descriptors . since the matrix descriptor is invariant to rigid transform and scaling , the distance function in eq . ( 4 ) is also invariant to rigid transform and scaling . similar to 3d object &# 39 ; s matrix descriptor , the 2d objects &# 39 ; matrix descriptor is also invariant to rotation , symmetric transform , and mirror transform . 2 5 it is clear that the distance functions for 2d / 3d objects in eq . 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