## 1 Introduction  
Quickly measuring three-dimensional (3D) surfaces with small 3D structures is a challenge for industrial measurement technology. For 100% inline-inspection of components, the measuring system has to keep up with the production cycle. Because of its great resolution and high speed, digital holography is a well-established technology for nondestructive testing in industrial applications [1; 2; 3; 4; 5; 6; 7]. Millions of data points can be acquired in a fraction of a second, which allows measuring cycles of multiple Hz.  
Lateral resolution of holographic sensors is limited by the pixel pitch of the imaging device and designed magnification. With the use of a lens, it can be both improved and adapted. With respect to axial resolution and unambiguity, the wavelength of the laser and the dynamic range of the imaging device are limiting factors. In addition, limited depth-of-focus is a challenge addressed by our algorithm. Spatial unwrapping allows to extend the measuring range, but often fails (e.g., at steep edges and deep holes) [8]. Using digital multiwavelength holography, speckle fields are captured at identical boundary conditions but at different wavelengths. This allows a numerical generation of phase maps of an artificial wavelength. These generated synthetic wavefronts overcome both restrictions [9; 10; 11; 12; 13; 14; 15; 16]. As an alternative to the additional expensive lasers and further appropriate longer synthetic wavelengths, the axial measurement range in digital holography can be extended using shape-from-focus algorithms for focus detection, applied on amplitude or phase images. Promising approaches have been highly investigated in the past (e.g., amplitude stack [17], complex ratio [18], correlation coefficient [19], energy conservation [19; 20], entropy [19], Fourier spectral [19; 20; 21; 22], Gini index [23], gradient [22], Kirchhoff [24], L1 norm [23; 25], Laplacian [21; 22; 26], multispectral images [19; 27], self-entropy [28], speckle phase decorrelation [29], Tamura coefficient [30; 31] variance [19; 21; 26; 32; 33] Klicken oder tippen Sie hier, um Text einzugeben, quadratic deformation of spatial coordinates [34], cubic phase plate [35], amplitude modulus [36], or numerical axicon transformation [37]).  
We present a shape-from-focus algorithm based on phase noise and use its output as a rough estimate of the object's surface shape, which allows us to extend the measurement range far beyond the limits of synthetic wavelengths and geometrical depth-of-focus. This algorithm has been previously introduced by us in 2017 [38], using a maximum filter in combination with a plane fit for height map generation. In this paper, we present a generalized improvement using a pixel wise Gaussian fit.  
The performance of our method is demonstrated using a gear test object at steep slopes and a step object with gaps beyond the measuring unambiguous range. Even at steep slopes, we are able to numerically compute an all-in-focus image of the gear flank beyond the measurement range of the synthetic wavelengths being used. In order to save computing time, data processing is mostly executed on the graphics processing unit (GPU).