TWI874727B - Method, machine-readable storage medium, and system for optimization-based image processing for metrology - Google Patents

Abstract

One or more images of a device feature are acquired using an imaging tool. A geometrical shape is defined encompassing the relevant pixels of each image, where the geometrical shape is represented in terms of one or more parameters. A cost function is defined whose variables comprise the one or more parameters of the geometrical shape. For each image, numerical optimization is applied to obtain optimal values of the one or more parameters for which the cost function is minimized. The optimal values of the one or more parameters are reported as metrology data pertaining to the device feature.

Description

Method, machine-readable storage medium and system for measurement based on optimization of image processing

Embodiments of the present disclosure generally relate to measuring fine features in devices on semiconductor wafers, and more particularly to obtaining accurate metrology data by using mathematically optimized image processing.

The manufacturing process of semiconductor integrated circuits requires high-resolution measurement of fine features for accurate metrology. Metrology data is often used to tune process parameters to improve manufacturing yield and uniformity. Acquiring high-resolution images and measuring dimensions (including critical dimensions, CD) directly from images is one way to generate metrology data. However, direct measurement is negatively affected by noise, which can be image noise inherent in the original image, measurement noise (e.g., image artifacts that are not present in the original imaged object but introduced by limitations of the imaging equipment), and/or other local artifacts (e.g., local residues or debris).

Image processing techniques are used to improve measurement accuracy. One such image processing technique is edge detection. In edge detection techniques, a smaller area of pixels in the original image is contained by the outline around the detected edge. However, edge detection techniques are particularly susceptible to image noise and artifacts. The present disclosure proposes a method for obtaining a measurement value that is a result of a numerical optimization problem and is more robust to noise.

The following is a simplified description of the disclosure to provide a basic understanding of certain aspects of the disclosure. This description is not an extensive overview of the disclosure. This description is neither intended to identify key or critical elements of the disclosure, nor to describe any scope of a specific implementation of the disclosure or any scope of the scope of the patent application. Its sole purpose is to present some concepts of the disclosure in a simplified form as a prelude to the more detailed description that is presented later.

In one aspect of the disclosure, an imaging tool is used to obtain one or more images of a device characteristic. The imaging tool may be an optical, electron beam, or X-ray based imaging tool, or any other imaging technique for obtaining images. A geometric shape is defined to encompass the relevant pixels of each image, wherein the geometric shape is represented according to one or more parameters. A cost function is defined whose variables include one or more parameters of the geometric shape. For each image, numerical optimization is applied to obtain an optimal value of the one or more parameters in which the cost function is minimized. The optimal value of the one or more parameters is reported as metrology data related to the device characteristic.

The geometric shape may be an ellipse, wherein the one or more parameters representing the ellipse include the diameter of the major axis of the ellipse, the diameter of the minor axis of the ellipse, the coordinates of the center of the ellipse, and the angular orientation of the ellipse. When a brighter background with dark pixels in the center represents a feature image, a single ellipse may be sufficient. The dark pixels may be encompassed by the ellipse outline.

In another aspect, the cost function can be adjusted so that numerical optimization of one or more parameters produces an elliptical ring that includes a ring of relatively bright pixels in a relatively dark background in an image with a low signal-to-noise ratio.

In yet another aspect, a device feature may include a three-dimensional (3D) hole having a top opening and a bottom surface that is not directly imageable by an imaging tool due to obstruction of a sloping sidewall connecting the top opening and the bottom surface of the hole. A top ellipse and a bottom ellipse are defined to include a first set of pixels representing the top opening of the hole and a second set of pixels representing the bottom surface of the hole, respectively. Defining the bottom ellipse in conjunction with known (i.e., a priori measured) dimensions compensates for the bottom surface not being directly imageable due to obstruction of the sloping sidewall. A cost function is tailored so that a numerical optimization produces an offset value between the top ellipse and the bottom ellipse.

Embodiments of the present disclosure are directed to a novel method for geometric image measurement based on numerical optimization that is robust to high noise levels or other types of image artifacts present in the original image. Image artifacts may be introduced by limitations of the imaging equipment, local debris or residues, or other inherent characteristics of the imaged device (e.g., blurring features).

One objective achieved by the present disclosure is to generate metrology data for fine features of electronic devices in a non-destructive manner using images obtained from various imaging tools, including but not limited to electron beam (e-beam) inspection tools (e.g., scanning electron microscope (SEM)), optical imaging tools, X-ray based imaging tools, etc. The electronic devices can be advanced semiconductor devices formed on a wafer. The 3D features can have lateral dimensions ranging from a few nanometers to tens or hundreds of nanometers. Some semiconductor devices can have fine features that have not only tight lateral dimensions but also high aspect ratios (HAR). However, the present disclosure is not limited to any particular lateral dimensions or any particular aspect ratio. Illustrative examples of imaged device features include, but are not limited to, channel holes, slits, trenches, etc. Specific examples of high depth-width features include circular memory holes in 3D NAND memory devices. One skilled in the art can extrapolate the application of the disclosed technology to any other geometric shapes. Examples of other geometric shapes include trenches, such as trenches used for shallow trench isolation of transistors. The 3D feature can be part of an isolation structure or an array of similar features.

Device features should be well characterized using detailed metrology to enable tuning of process parameters. For example, as a process (e.g., an etching process or a deposition process) progresses, the aspect ratio of a feature changes. As a specific illustration, in an etching process, the etch rate varies as the aspect ratio of the feature changes over time. Accurate characterization of device features enables effective tuning of etching process parameters. Current device characterization methods use electron beam/optical/X-ray images along vertical (or longitudinal) cross sections, and/or transmission electron microscopy (TEM) images. Such destructive imaging techniques typically provide only an image of a single planar cross-section (longitudinal cross-section) from which a limited number of device characterization metrics can be obtained, which is not suitable for high-volume manufacturing (HVM). The present disclosure addresses these and other shortcomings of current methods by using mathematical optimization to measure device features from top-down imaging without having to destroy the wafer to expose the longitudinal cross-section.

Advantages of the method include, but are not limited to, robustness to noise and image artifacts, flexibility in choosing measurement parameters, and flexibility in defining a cost function associated with the optimal value of a desired measurement metric.

As described in the Background section, one existing method of providing metrology data in a non-destructive manner is to obtain a top-down image and use image processing techniques based on edge detection. FIG. 1A illustrates an image 100 of a top view of a device feature having a circular geometry. FIG. 1B illustrates three regions of the image 100 identified via image processing (labeled A, B, and C), and individual outlines (dashed outlines) are superimposed on the original image along the detected edges of each of those smaller regions. FIG. 1B represents state-of-the-art image processing-based metrology. In stark contrast to currently used edge detection-based methods, the present disclosure discloses using a single geometric outline 110 (as shown in FIG. 1C ) to encompass the entire region D of relevant pixels of the image 100 and numerically finding the optimal parameters of the geometric outline 110. This method works well when the background 120 is relatively bright and the features are represented by relatively dark pixels within the region D.

The geometric outline 110 may be in the shape of an ellipse 200, as shown in FIG. 2. The parameters defining the ellipse 200 include the length "a" of the major axis 210, the length "b" of the minor axis 220, the center coordinates (x, y), and the direction of the ellipse, such as the angle θ between the major axis 210 and the horizontal axis 230. The cost function may be defined based on the above parameters of the ellipse. For example, the cost function is defined as: cast ( x,y,a,b,θ )= Values ( x,y,a,b,θ )- λ Area ( a,b )...(Equation 1) where a=the diameter of the major axis of the ellipse, b=the diameter of the minor axis of the ellipse, (x,y) are the coordinates of the center of the ellipse, θ is the angle between the horizontal axis and the major axis of the ellipse, indicating the angular direction of the ellipse, area=the area of the ellipse; and λ is a tuning parameter used to optimize the cost function.

The cost function has two terms: the first term is associated with the gray value of the pixel , and the second term related to the area of the ellipse . The tuning parameter λ is a coefficient that controls the trade-off between the first and second terms in order to minimize the final cost function. For example, when the first term becomes smaller and the second term becomes larger, the value of the cost function decreases. As the area of the ellipse becomes larger, the second term becomes larger. The first term becomes smaller when the gray value of the pixel decreases, that is, when the included pixels are much darker than the brighter background surrounding the relevant pixel. This type of cost function is best suited for the first case of images with a relatively high signal-to-noise ratio.

In the second case, the original image may be very noisy, that is, the signal-to-noise ratio within the entire elliptical area may not be the most ideal cost function. For example, as shown in Figure 3A, the original image 300 has a darker background, but there is an approximately annular area 310 with brighter pixels within the image. For this type of situation, it is a better approach to define two ellipses, an outer ellipse 320 and an inner ellipse 330, rather than one ellipse, as shown in Figure 3B. The two ellipses 320 and 330 together define an elliptical ring that surrounds the annular area 310 with brighter pixels. The cost function is customized so that the value optimizes one or more parameters that produce the elliptical ring. The parameters of the elliptical rings are then used as metrology data to tune the existing process. The parameters of the elliptical rings may include the ring's position (center of the two ellipses), width (i.e., the difference between the radius of the outer ellipse and the radius of the inner ellipse), and directivity (orientation, i.e., the angle of the major axis relative to the horizontal axis).

In the third case, the disclosed numerical optimization techniques and a priori measurement data incorporated into the cost function can be used very effectively. This is particularly useful for exemplary images of an etched hole (which may be a high aspect ratio 3D structure) having slanted sidewalls 440A and 440B, as shown in the longitudinal view 400 shown in FIG. 4A. The hole is etched within a substrate body 420. The hole has a top opening 450 and a bottom surface 460 connected by sidewalls 440A and 440B, which may deviate from ideal parallel sidewalls 430A and 430B. If the etching process is ideal and the depth of the hole increases without effect as the etching process proceeds, parallel sidewalls will result.

FIG. 4B illustrates a top view 410 of the imaged aperture shown in FIG. 4A in side view. Top ellipse 480 encompasses pixels representing top opening 450, while bottom ellipse 490 encompasses pixels representing bottom surface 460. Note that here too, background 470 is relatively brighter than the center region of darker pixels representing the aperture (similar to FIG. 1C ). Due to the tilted sidewall 440B, a portion of bottom surface 460 is obscured from the line of sight of the imaging tool acquiring the top-down image. However, the dimensions of the bottom surface are known based on prior measurements (which may be destructive or non-destructive). Thus, the parameters of bottom ellipse 490 may depend in part on prior knowledge of the dimensions of the obscured feature. The cost function is tailored so that the numerical optimization produces an offset value 495 between the center of the top ellipse 480 and the bottom ellipse 490. The offset value 495 can be an important metrology data based on which the process parameters are tuned. Therefore, despite the partial obstruction of the bottom surface, a reliable offset value can be produced by incorporating prior knowledge into the numerical optimization technique described herein.

FIG. 5 is a flow chart of an exemplary method 500 for generating metrology data based on numerical optimization according to some embodiments of the present disclosure. The method 500 may be performed by processing logic that may include hardware (e.g., a processing device, a circuit system, dedicated logic, programmable logic, microcode, hardware of a device, integrated circuits, etc.), software (e.g., instructions running or executed on a processing device), or a combination thereof. Although shown in a particular sequence or order, the order of processes in the method 500 or other methods described herein with illustrative flow charts may be modified unless otherwise stated. Therefore, the illustrated embodiments should be understood as examples only, and the illustrated processes may be performed in a different order, and some processes may be performed in parallel. Additionally, one or more processes may be omitted in various embodiments. Therefore, not all processes are required in every embodiment. Other process flows are possible.

In method 500, at block 510, imaging tool parameters may be adjusted prior to acquiring an image to maximize signal-to-noise ratio, as appropriate. As described above, the imaging tool may be electron beam based, optical based, or X-ray based. The scope of the present disclosure is in no way limited by the type of imaging tool used.

At block 520, an image is acquired by an imaging tool of the device feature to be measured. For example, the diameter of a hole may be a critical dimension (CD) to be measured to generate metrology data. Typically, more than one image of the device feature is acquired.

At block 530, the geometric outline of the measurement area and parameters of the outline are defined by the processing device. For example, if an ellipse is to encompass the pixels of the imaging device feature, the parameters of the ellipse are defined, such as the major axis, minor axis, center coordinates, and directionality of the ellipse, as described in FIG.

At block 540, a cost function is defined based on the parameters of the selected geometric profile. An example of a cost function is previously given in Equation 1. The first term of the equation may be the sum of all grayscale values of the associated pixels. The tuning parameter may be associated with a predetermined threshold grayscale value, such as the average grayscale value of all pixels in the original image, including the associated pixels representing device features and background pixels representing the surrounding substrate. The cost function may be customized to represent the average pixel value inside an ellipse (e.g., FIG. 1C ), the pixel value inside an elliptical ring (e.g., FIG. 3B ), the displacement or offset between two elliptical profiles (e.g., FIG. 4B ), or any other cost function that is a convenient metrology metric for process control.

At block 550, the processing device applies a numerical optimization technique to find the parameter values that minimize the resulting cost function. One example of numerical optimization is based on the Nelder-Mead method, but those skilled in the art will understand that the scope of the present disclosure is not limited to the use of any particular optimization technique.

At block 560, the optimal values of the parameters that minimize the cost function are provided as outputs. The outputs may be reported as input data to a metrology-based process control tool.

FIG. 6 illustrates a machine of a computer system 600 in which a set of instructions for causing the machine to perform any one or more of the methods discussed herein may be executed. In alternative implementations, the machine may be connected (e.g., networked) to other machines in a local area network (LAN), an intranet, an inter-enterprise network, or the Internet. The machine may operate as a server or client machine in a client-server network environment, or as a peer machine in a peer-to-peer (distributed) network environment, or as a server or client machine in a cloud computing infrastructure or environment.

The machine may be a personal computer (PC), a tablet, a set-top box (STB), a network device, a server, a network router, a switch or a bridge, or any machine capable of executing (sequentially or otherwise) a set of instructions that specify actions to be taken by the machine. Furthermore, although a single machine is illustrated, the term "machine" should also be taken to include any collection of machines that individually or collectively execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein.

The exemplary computer system 600 includes a processing device 602, a main memory 604 (e.g., read-only memory (ROM), flash memory, dynamic random access memory (DRAM) (such as synchronous DRAM (SDRAM)), etc.), a static memory 606 (e.g., flash memory, static random access memory (SRAM), etc.), and a data storage device 616, all of which communicate with each other via a bus 608.

Processing device 602 represents one or more general-purpose processing devices, such as a microprocessor, a central processing unit, etc. More specifically, the processing device can be a complex instruction set computing (CISC) microprocessor, a reduced instruction set computing (RISC) microprocessor, a very long instruction word (VLIW) microprocessor, a processor implementing other instruction sets, or a processor implementing a combination of instruction sets. Processing device 602 can also be one or more special-purpose processing devices, such as an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a digital signal processor (DSP), a network processor, etc. The processing device 602 is configured to execute instructions for performing the operations and steps discussed herein.

The computer system 600 may further include a network interface device 622 for communicating on the network 618. The computer system 600 may also include a video display unit 610 (e.g., a liquid crystal display (LCD) or a cathode ray tube (CRT)), an alphanumeric input device 612 (e.g., a keyboard), a cursor control device 614 (e.g., a mouse or a touch pad), a signal generating device 620 (e.g., a speaker), a graphics processing unit (not shown), a video processing unit (not shown), and an audio processing unit (not shown).

The data storage device 616 may include a machine-readable storage medium 624 (also referred to as a computer-readable medium) on which one or more sets of instructions or software implementing any one or more of the methods or functions described herein are stored. The instructions may also reside completely or at least partially in the main memory 604 and/or in the processing device 602 during their execution by the computer system 600, the main memory 604 and the processing device 602 also constituting machine-readable storage media.

In one implementation, the instructions include instructions for implementing functions corresponding to the height difference determination. Although the machine-readable storage medium 624 is illustrated as a single medium in the exemplary implementation, the term "machine-readable storage medium" should be construed to include a single medium or multiple media (e.g., a centralized or distributed database, and/or associated cache memory and server) storing one or more sets of instructions. The term "machine-readable storage medium" should also be construed to include any medium capable of storing or encoding a set of instructions executed by a machine and causing the machine to perform any of the one or more methods of the present disclosure. The term "machine-readable storage medium" should accordingly be construed to include, but not be limited to, solid-state memory and optical and magnetic media.

Some portions of the foregoing detailed description have been presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. Such algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to those skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of operations leading to a desired result. The operations are those requiring physical manipulation of physical quantities. Usually, though not necessarily, such quantities take the form of electrical or magnetic signals capable of being stored, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.

However, it should be considered that all such and similar terms are associated with the appropriate physical quantities and are merely convenient labels applied to such quantities. Unless otherwise clear from the above discussion, it should be understood that throughout the description, discussions using terms such as "obtain" or "associated" or "perform" or "produce" represent actions or processes of a computer system or similar electronic computing device that operates and converts data represented as physical (electronic) quantities in the computer system's registers and memories and converts it into other data similarly represented as physical quantities in the computer system's memories or registers or other such information storage devices.

The present disclosure also relates to an apparatus for performing the operations described herein. The apparatus may be specially constructed for the intended purpose, or it may comprise a general purpose computer selectively activated or reconfigured by a computer program stored in the computer. The computer program may be stored in a computer readable storage medium such as, but not limited to, any type of disk, including floppy disks, optical disks, CD-ROMs and magneto-optical disks, read-only memory (ROM), random access memory (RAM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), magnetic or optical cards, or any type of medium suitable for storing electronic instructions, each of which is coupled to a computer system bus.

The algorithms and displays presented herein are not intrinsically related to any particular computer or other device. Various general purpose systems may be used with programs according to the teachings herein, or it may prove convenient to construct more specialized devices to perform the methods. Structures for various such systems will appear in the following description. In addition, the present disclosure is not described with reference to any particular programming language. It should be understood that a variety of programming languages may be used to implement the teachings of the present disclosure as described.

The present disclosure may be provided as a computer program product or software, which may include a machine-readable medium having stored thereon instructions that may be used to program a computer system (or other electronic device) to perform a process according to the present disclosure. Machine-readable media include any mechanism for storing information in a form readable by a machine (e.g., a computer). For example, machine-readable (e.g., computer-readable) media include machine (e.g., computer) readable storage media such as read-only memory ("ROM"), random access memory ("RAM"), disk storage media, optical storage media, flash memory devices, etc.

In the foregoing specification, the embodiments of the present disclosure have been described with reference to specific exemplary implementations of the present disclosure. It will be apparent that various modifications may be made to the present disclosure without departing from the broader spirit and scope of the embodiments of the present disclosure as set forth in the claims below. Accordingly, the specification and drawings are to be regarded as illustrative rather than restrictive.

100: Image 120: Background 200: Ellipse 210: Long axis 220: Short axis 230: Horizontal axis 300: Original image 310: Annular area 320: Outer ellipse 330: Inner ellipse 400: Vertical view

410: Top view of the imaging hole

420: Substrate body

430A: Parallel side walls

430B: Parallel side walls

440A: Side wall

440B: Side wall

450: Top opening

460: Bottom surface

470: Background

480: Top oval

490: Bottom ellipse

495:Offset value

500:Methods

520: Operation

530: Operation

540: Operation

550: Operation

560: Operation

600: Computer system

602: Processing device

604: Main memory

606: Static Memory

608:Bus

610: Video display unit

612: Alphanumeric input device 614: Cursor control device 616: Data storage device 618: Network 620: Signal generating device 622: Network interface device 624: Machine readable storage medium (x, y): Center coordinates θ: Angle

The present disclosure will be more fully understood from the detailed description given below and from the accompanying drawings of various embodiments of the present disclosure.

FIG. 1A shows a top view of an original image of a device feature (memory hole).

FIG. 1B shows an elliptical outline superimposed on the original image of FIG. 1A for a conventional edge-based detection technique.

FIG. 1C illustrates a preferred ellipse superimposed on the original image of FIG. 1A according to an embodiment of the present disclosure.

FIG. 2 illustrates parameters for constructing an optimal ellipse of a cost function according to an embodiment of the present disclosure.

FIG. 3A illustrates the original image of the feature, which shows a nearly bright elliptical ring of a certain width against a relatively dark background.

FIG. 3B illustrates the construction of a cost function customized for bright elliptical rings with a dark background according to an embodiment of the present disclosure.

FIG. 4A illustrates a schematic longitudinal cross-sectional side view of a 3D memory hole with non-uniform sidewall tilt.

FIG. 4B illustrates a preferred ellipse superimposed on the top view image of the memory hole schematically shown in its entirety in FIG. 4A according to an embodiment of the present disclosure.

FIG. 5 illustrates a flow chart describing an exemplary method for metrology-based optimization-based image processing according to an embodiment of the present disclosure.

FIG. 6 illustrates an exemplary computer system in which a set of instructions for performing any one or more of the methodologies discussed herein may be executed.

Domestic storage information (please note in the order of storage institution, date, and number) None Foreign storage information (please note in the order of storage country, institution, date, and number) None

320: Outer ellipse

330: Inner ellipse

Claims (20)

A method for providing metrological data by optimization-based image processing performed by a processing device, the method comprising the following steps: using an imaging tool to obtain one or more images of a device feature, wherein the one or more images are top-down images; defining a geometric shape covering relevant pixels of each image of the one or more images, wherein the geometric shape is represented according to one or more parameters; defining a cost function whose variables include the one or more parameters of the geometric shape; for each image, applying numerical optimization to obtain optimal values of the one or more parameters in which a value of the cost function is minimized; and providing the optimal values of the one or more parameters as metrological data about the device feature. The method as described in claim 1 further comprises the following steps: before acquiring the one or more images, tuning imaging tool parameters to maximize the signal-to-noise ratio of the one or more images. A method as described in claim 1, wherein the geometric shape comprises an ellipse, wherein the one or more parameters representing the ellipse comprise a major axis diameter of the ellipse, a minor axis diameter of the ellipse, the coordinates of a center of the ellipse, and the angular direction of the ellipse. A method as described in claim 3, wherein the cost function is defined as: cost ( x,y,a,b,θ ) = Values ( x,y,a,b,θ ) - λ Area ( a,b ) wherein a = the diameter of the major axis of the ellipse, b = the diameter of the minor axis of the ellipse, (x,y) are the coordinates of the center of the ellipse, θ is an angle between a horizontal axis and the major axis of the ellipse, representing an angular direction of the ellipse, area = the area of the ellipse; and λ is a tuning parameter used to optimize the cost function. A method as described in claim 4, wherein the tuning parameter λ controls the tradeoff between a first term of the cost function ( Values ( x,y,a,b,θ )) and a second term of the cost function ( λ Area ( a,b )). The method of claim 4, wherein the optimal values of the parameters of the ellipse for which the cost function is minimized represent the largest and darkest ellipse that encompasses the relevant pixels in a relatively bright background. The method as described in claim 3 further comprises the following steps: detecting an annulus having relatively bright pixels in a relatively dark background in an image having a low signal-to-noise ratio; defining an inner ellipse and an outer ellipse that together form an elliptical ring, the elliptical ring covering the ring of relatively bright pixels; and adjusting the cost function so that the value optimizes one or more parameters that generate the elliptical ring. The method as described in claim 3, wherein the device features a hole having a top opening and a bottom surface, the bottom surface being partially obscured because it cannot be directly imaged by the imaging tool, wherein the top opening and the bottom surface are connected by an inclined side wall. The method as described in claim 8 further comprises the steps of: obtaining known dimensions of the bottom surface of the hole from previous measurements; defining a top ellipse and a bottom ellipse respectively including a first set of pixels representing the top opening and a second set of pixels representing the bottom surface, wherein the bottom ellipse combines the known dimensions of the bottom surface obtained from previous measurements; and customizing the cost function so that the value optimizes an offset value between the top ellipse and the bottom ellipse. The method of claim 9, wherein defining the bottom ellipse in conjunction with the known dimensions compensates for the bottom surface that cannot be directly imaged due to being obscured by the inclined side walls. A non-transient machine-readable storage medium storing instructions that, when executed, cause a processing device to perform operations comprising: defining a geometric shape for pixels associated with each of one or more images comprising a device feature obtained using an imaging tool, wherein the one or more images are top-down images, and wherein the geometric shape is represented according to one or more parameters; defining a cost function whose variables include the one or more parameters of the geometric shape; For each image, applying numerical optimization to obtain optimal values of the one or more parameters where a value of the cost function is minimized; and providing the optimal values of the one or more parameters as metric data regarding the device feature. A non-transitory machine-readable storage medium as described in claim 11, wherein the geometric shape comprises an ellipse, wherein the one or more parameters representing the ellipse comprise a major axis diameter of the ellipse, a minor axis diameter of the ellipse, coordinates of a center of the ellipse, and an angular orientation of the ellipse. A non-transient machine-readable storage medium as described in claim 12, wherein the cost function is defined as: cost ( x,y,a,b,θ ) = Values ( x,y,a,b,θ ) - λ Area ( a,b ) wherein a = the diameter of the major axis of the ellipse, b = the diameter of the minor axis of the ellipse, (x,y) are the coordinates of the center of the ellipse, θ is the angle between a horizontal axis and the major axis of the ellipse, indicating the angular direction of the ellipse, area = the area of the ellipse; and λ is a tuning parameter used to optimize the cost function. The non-transient machine-readable storage medium of claim 12, wherein the optimal values of the parameters of the ellipse for which the cost function is minimized represent the largest and darkest ellipse that encompasses the associated pixel against a relatively bright background. The non-transient machine-readable storage medium of claim 12, wherein the processing device further performs: Detecting an annular hole having relatively bright pixels in a relatively dark background in an image having a low signal-to-noise ratio; defining an inner ellipse and an outer ellipse that together form an elliptical ring, the elliptical ring encompassing the ring of relatively bright pixels; and customizing the cost function so that the value optimizes one or more parameters that generate the elliptical ring. The non-transient machine-readable storage medium of claim 12, wherein the processing device further performs: obtaining known dimensions of a bottom surface of a device feature from previous measurements, wherein the device feature comprises a hole having a top opening and the bottom surface, the bottom surface being partially obscured because it cannot be directly imaged by the imaging tool, wherein the top opening and the bottom surface are connected by an inclined side wall; defining a top ellipse and a bottom ellipse respectively comprising a first set of pixels representing the top opening and a second set of pixels representing the bottom surface, wherein the bottom ellipse combines the known dimensions of the bottom surface obtained from previous measurements; and customizing the cost function so that the value optimizes an offset value between the top ellipse and the bottom ellipse. A system for measurement-based optimization image processing includes a memory and a processing device coupled to the memory, wherein the processing device performs the following operations: obtaining one or more images of a device feature, wherein the one or more images are top-down images; defining a geometric shape including relevant pixels of each image of the one or more images, wherein the geometric shape is represented according to one or more parameters; defining a cost function whose variables include the one or more parameters of the geometric shape; for each image, applying numerical optimization to obtain optimal values of the one or more parameters in which a value of the cost function is minimized; and providing the optimal values of the one or more parameters as measurement data about the device feature. The system as described in claim 17 further comprises: an imaging tool that acquires the one or more images and sends the one or more images to the processing device, wherein before acquiring the one or more images, imaging tool parameters are tuned to maximize the signal-to-noise ratio of the one or more parameters. A system as described in claim 17, wherein the geometric shape comprises an ellipse, wherein the one or more parameters representing the ellipse comprise a major axis diameter of the ellipse, a minor axis diameter of the ellipse, the coordinates of a center of the ellipse, and the angular orientation of the ellipse. The system of claim 17, wherein the cost function can be adjusted so that the numerical optimization produces one or more parameters of an elliptical ring that includes a ring of bright pixels in a relatively dark background in an image with a low signal-to-noise ratio.