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Due to limited budget, I have been thinking I need to save money and not buy another lens. However, I also want to shoot wide angle. So I think whether there are differences between using wide angle lenses and 50mm to shoot then using panorama to stitch them together?

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Those are both choices to produce an image with a wide angle-of-view. The result will not be exactly the same though.

Since you have an interchangeable lens camera, you have full manual controls which means you can lock exposure, focus and white-balance so that you can get consistent shots to stitch together. Once you have stitched images together, you will have a much higher-resolution image than you would with a single shot. This can be a huge advantage and allow much larger print sizes or lower noise when it is printed at less than the maximum possible size.

There are some disadvantages though:

  • Anything that moves is problematic. If something moves in the overlap area, then it will cause some stitching problems. Depending on the case, the results may or may not be usable. Some software have ghost removal tools which help but do not work 100% of the time. Large moving object such as trucks and cars often cause problems.
  • To get a prefect set of images for stitching you must rotate the lens around its nodal point. This requires a tripod and panoramic head matched to your lens or a configurable one which is much more bulky and expensive. Doing it by hand is possible and often done but the result may show parallax problems. This occurs when you have foreground objects.
  • Stitching a panorama takes time and processing power. You will not see your results right away which means having to return home or to the hotel before knowing if the sequence worked. This is honestly the most troublesome part. Even with panoramic head and tripod, I sometimes get a set of images which software is unable to stitch correctly. Subject motion and user error are the most common source of these issues. Knocking the focus ring between images for example happened recently to me.
  • It takes time on the ground too. When shooting panoramas, I end up with much fewer images since the time to setup is relatively high. You have to position the tripod, set up the head, mount the camera. You also have to do your own metering because the optimal exposure rarely is the same for each individual frame as it is for the whole. Same with WB.
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A wide angle lens covers more in both width and height. If you want to have the same result by stitching the you must not only stitch pictures 'next to each other' but also 'above each other'.

The big problems with stitching are:

  • Light. If you go from or to a light source then the colours can be different on the left and on the right.
  • Moving objects. If objects (persons/animals/cars) move into the same direction as you take your pictures then they can appear multiple times on the result.
  • Objects in the front can make it difficult to give a good result.
  • Straight lines can become curved.

You can try to 'fix' these problems but it will take you a lot of effort and time. You cannot have it all :-)

All these problems you do not have with a wide angle lens.

  • Ususally when using my normal lens to grab a few stitchable hand-held shots for a wider view, I use portrait orientation to avoid having to shoot more than one row. It seems easier to me than trying to pan in multiple dimensions while keeping the camera position consistent. – junkyardsparkle Dec 3 '16 at 20:43
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Depending on your shooting style and subject matter, I will make the argument for stitching together images using a narrow(-er, -ish) lens.

Aside from typical high-DoF panoramas (landscapes, architecture, etc.), "bokehrama" stitching of images, the Brenizer Method (named after wedding photography Ryan Brenizer), is just that: stitching together many shots from a normal or portrait focal length lens (50mm–135mm or so, relative to 35mm full-frame cameras), usually with a wide aperture. It achieves the look of wide field of view combined with narrow depth of field, that could only otherwise be produced by larger format cameras with very fast (large aperture) wide angle lenses that either don't exist or are too expensive to buy for most photographers.

Differences between stitching images from a narrower lens and using a wider angle lens include:

  • Depth of Field: Stitching allows you create images with shallow depths of field that cannot be created by any lens. This is the hallmark of the Brenizer Method. For example, let's say you're taking a portrait of a person 10 ft away, and you want it framed like it was shot with a 24mm lens. Using your trusty 50mm lens at f/2.2 at that distance, and stitching images to create the same FoV as a 24mm lens, you would have the same depth of field as a hypothetical 24mm lens @ f/0.5 (!) at the same distance. (I say hypothetical, because no such lens has ever, nor will ever, exist).

    Alternately, instead of shooting your 50mm wide open, perhaps you want to stop it down to f/5.6 because it's sharpness sweet spot is around f/5.6. You're not going for thin DoF as much as you want absolute sharpness. You still have slightly more DoF with stitched 50mm f/5.6 than you can get with a 24mm f/1.4 (the actual aperture for identical DoF at 24mm would need to be f/1.28). Stitching your decent "Nifty Fifty" lets you simulate a $850–$2000 24mm f/1.4 lens (for some values of "simulate").

  • Curved Plane of Focus: With a single lens, the plane of focus is perpendicular to the lens axis. Aim a camera perfectly level, perpendicular to a wall; all of the wall that is in the field of view will be in focus, even at the widest possible aperture (ignoring any slight field curvature that all lenses exhibit to lesser or greater degrees).

    However, when you take multiple images of that wall by rotating the camera laterally or vertically, without changing focus, then the wall falls out of focus the further you point the camera away from perpendicular to the wall.

    Example: perhaps you want to take a shot of a group of people. With a wide angle lens, framing a single shot so everybody is in frame, you would have them line up in a straight line that is perpendicular to the camera. However, the people at the ends would usually look distorted — they appear wider/fatter than they really are, and their heads and feet are stretched towards the corners of the frame (depending on how close you are to the group of people, i.e., how much of the frame the group fills).

    Following the same example, but using a longer lens and stitching multiple shots to achieve the same field of view, you would have the people line up in an arc, equidistant from you, to keep them in focus.

    Note(†): When I say "curved plane" of focus, obviously a plane can't be curved. Perhaps better to call it a surface of focus. Also, the surface of focus won't be curved; rather, it will be piecewise planar (either a segmented cylinder, or segmented sphere sort of like the surface of a soccer ball; a section of an irregular polyhedron).

    Also note: The "field curvature" of a Brenizer method image is not the same as Petzval field curvature. A Brenizer method–image won't produce the "swirly bokeh" of a Petzval lens. But it will produce shallow DoF with a curved surface of focus.

  • Image Size: Obviously, stitching produces a larger image, creating a look that can only otherwise be achieved by using a large film or sensor format. Most photographers don't have the means or access to medium or large format equipment that is required to produce large, high-resolution prints. Stitching is the other way to achieve this.

  • Available Shots: Also obviously, there are several shots that are impossible, or incredibly infeasible, to achieve via stitching. Because each image is literally a snapshot of a point in time, a sequence of images is a sequence of different points in time. Anything that changes over time (backgrounds, movement, etc.) changes the scene. If you are trying to recreate a single snapshot-point-in-time, nothing in the scene can move. For instance, stitching looks bad when the clouds don't line up because they are moving too fast, or when ocean waves don't match. The only thing you can do there is some creative Photoshop work (which may or may not suit your taste or style), or long-exposure multiply-stitched images (which creates additional problems if you're trying to convince a subject to stay still for the entire image sequence).

    On the other hand, using the examples in previous points above, there are images that are just not achievable or practical when taking a single shot with a wide angle lens.

  • Time Penalty: The flipside of the coin in the previous point is that taking multiple images incurs a time penalty while taking the image. Keeping a single subject or couple still while taking a Brenizer method sequence is doable, perhaps requires a do-over or two if the subject(s) move, or there are unexpected foreground or background changes through the scene. Normal photography lets you take several shots per minute. You can still take as many frames when doing panoramas, but obviously, batches of those frames are all in service of a single final image. So your keeper rate will be lower.

    And of course, there's the obvious time penalty in post-processing. You still have to do the normal retouching, color balancing, etc., for each final image you produce as you normally would. But arranging, aligning, examining, re-aligning, etc., the images in a sequence can take quite a bit of time.

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